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A partial digestion of the HRT counterexample

6 August, 2026 in expository, math.CA | Tags: , | by | 9 comments

A function {f(t) \in L^2({\bf R})} of one variable can be translated in space by a spatial shift {x} to obtain a new function

\displaystyle \pi(x,0) f(t) := f(t-x),

and also modulated in frequency by a frequency shift {\omega} to obtain a new function

\displaystyle \pi(0,\omega) f(t) := e^{2\pi i \omega t} f(t).

One can compose these two operations to obtain a time-frequency shift:

\displaystyle \pi(x,\omega) f(t) := e^{2\pi i \omega t} f(t-x).

(This can be viewed of as a portion of the Weyl representation of the Heisenberg group, but we will not adopt a representation-theoretic perspective here.)

Some functions obey finite linear relations between their time-frequency shifts. For instance, a sinusoid {f(t) = A \sin(k t + \phi)} obeys the relation

\displaystyle \pi(\frac{\pi}{k},0) f + f = 0.

However, the Heil-Ramanathan-Topiwala (HRT) conjecture states that once one imposes some reasonable decay condition on {f}, no such relations exist:

Conjecture 1 (HRT conjecture) If {f \in L^2({\bf R})} is non-zero, then there is no relation of the form

\displaystyle c_1 \pi(z_1) f + \dots + c_n \pi(z_n) f = 0

for some distinct points {z_1,\dots,z_n \in {\bf R}^2} and some coefficients {c_1,\dots,c_n \in {\bf C}}, not all zero.

A special case of the HRT conjecture, which was also open, makes the additional assumption that {f} was Schwartz.

Many positive results towards this conjecture were known. I will mention only a few here. There is a result of Linnell that the conjecture is true if {z_1,\dots,z_n} lie in a translate of a discrete subgroup of {{\bf R}^2}; this (together with an argument handling the collinear case) establishes all cases where {n \leq 3}, and several partial results involving the {n=4} cases are also known. The conjecture is also known if {f} is decays at a suitably super-exponential rate, by work of Bownik and Speegle.

I was aware of this conjecture through various talks and conversations with colleagues, and even briefly tried my hand at it for a while, though not with particularly serious effort (or progress). It was thus a nice surprise to see that it has just been resolved by Faulhuber, Petersen, van Velthoven, and Voigtlaender, even in the Schwartz case:

Theorem 2 There exist complex numbers {c_1,\dots,c_{12}\in {\bf C}}, not all zero, distinct points {z_1,\dots,z_{12} \in {\bf R}^2}, and a non-zero Schwartz function {f_* \in \mathcal{S}({\bf R})} such that

\displaystyle c_1 \pi(z_1) f_* + \dots + c_{12} \pi(z_{12}) f_* = 0.

It is perhaps unsurprising in this current era that this result is AI-assisted. However, I think the authors have disclosed their AI use responsibly, with the final arguments written by hand with a readable overview of the argument, as well as proper discussion of methods, relation to past literature, and other independent numerical checks on the result.

The negative result lies only a little beyond the positive results: {n} is now increased to {12}, and all but one of the points {z_1,\dots,z_{12}} lie in (a translate of) a discrete subgroup of {{\bf R}^2} (in fact the explicit subgroup {{\bf Z} \times \frac{1}{2}{\bf Z}} is used). The functions constructed are smooth and rapidly decaying, but not analytic or super-exponentially decaying, which would start being in conflict with the known positive results.

In addition to AI being used to come up with the initial proof strategy, a more traditional numerical computation was used to verify one step of the argument.

I have not had the time to do a full digestion of the result, but (after reading the introduction, and using a little AI assistance of my own) I was able to understand the main ideas at a high level. The first few reductions are relatively standard. Setting {z_{12} = 0} and {c_{12} = -c_*}, one can view the problem as one of solving an eigenvalue problem

\displaystyle (c_1 \pi(z_1) + \dots + c_{11} \pi(z_{11})) f_* = c_* f_*.

The time-frequency shifts {z_1,\dots,z_{11}} are chosen to lie in a translate of the discrete subgroup {{\bf Z} \times \frac{1}{2}{\bf Z}} by a certain irrational shift {(\alpha, \beta/2)}. If one was working in the shifts of a standard lattice {{\bf Z} \times {\bf Z}}, it would be natural to work with the Zak transform of {f_*}, but it turns out that the approach does not quite work when doing this for topological reasons (relating to the fact that scalar quasiperiodic functions of mean zero are forced to have zeroes), and so the authors used the slightly denser lattice instead {{\bf Z} \times \frac{1}{2}{\bf Z}}, which relates to a vector-valued version of the Zak transform taking values in {{\bf C}^2} rather than {{\bf C}}. Applying this transform, the eigenvalue problem can be transformed by standard calculations to a “vector cocycle problem”

\displaystyle B_*(z) F_*(z - \tau) = c_* F_*(z), \ \ \ \ \ (1)

 where {F_* : {\bf R}^2 \rightarrow {\bf C}^2} is a non-zero smooth quasiperiodic vector-valued function, {\tau} is an irrational shift {\tau= (\alpha,\beta)}, and {B_*(z)} is a certain explicit {2 \times 2} matrix-valued function depending on the choices of {c_1,\dots,c_{11}}, {z_1,\dots,z_{11}}, and {\tau}.

How to solve this equation? The motivating scenario here is if the matrix function {B_*(z)} was replaced by a rank one function

\displaystyle B_0(z) = \chi(z) \chi(z-\tau)^*

for some smooth vector-valued function {\chi : {\bf R}^2 \rightarrow {\bf C}} of unit magnitude. Then one could solve the equation by taking {F_*(z) = \chi(z)} and {c_* = 1}. It is not possible to make the function {B_*} exactly of this form, but through some numerical computation and clever AI-assisted guesswork, the authors were able to find a choice of {c_1,\dots,c_{11}} and {z_1,\dots,z_{11}}, and {\tau} that made {B_*} approximately equal to a rank one function {B_0} of this form, in fact getting a uniform estimate

\displaystyle \sup_{z \in {\bf R}^2} \| B_*(z) - B_0(z) \|_{op} < \frac{1}{3}.

As it turns out, such an approximation is sufficient to run a contraction mapping argument to find a solution to a variant of (1), namely

\displaystyle B_*(z) v_*(z - \tau) = q_*(z) v_*(z)

for some smooth {v_* : {\bf R}^2 \rightarrow {\bf C}^2} and {q_* : {\bf R}^2 \rightarrow {\bf C}}. (Here it was important to get the operator norm bound below {\frac{1}{3}}; they are barely able to do this, with a numerically obtained bound of {0.333032}, though this bound might not be optimal.)

The main remaining obstacle is that the “eigenvalue function” {q_*(z)} is varying in the parameter {z} rather than constant. (This issue was, by the way, anticipated to some extent in previous work of Demeter, who observed that eigenfunctions of the almost Matthieu discrete Schrödinger operator gave a near-miss counterexample to the HRT conjecture, but with an eigenvalue that depended on an auxiliary phase shift parameter rather than constant.) However, if one was able to solve the scalar cocycle equation

\displaystyle q_*(z) h(z-\tau) = c_* h(z) \ \ \ \ \ (2)

for some smooth {h : {\bf R}^2 \rightarrow {\bf C}}, then one could solve the equation (1) by setting {F_*(z) = h(z) v_*(z)}. The approach to solve (2) is standard: take logarithms, apply a Fourier transform, and then divide out by the multiplier associated to the {\tau} shift. This can cause a well-known “small divisor” problem (which arises in various dynamical contexts, such as in the KAM theorem) if {\tau} behaves too much like a rational vector, but the standard resolution to this is to select a shift {\tau} that obeys good Diophantine approximation properties. For the purposes of numerics the authors selected an extremely concrete shift, namely

\displaystyle \tau = (2^{1/3} - 1, 2^{2/3} - 1)

but I get the impression that the exact choice here was not crucial for the argument, and that many other irrational algebraic numbers could have worked here.

A digestion of the Jacobian conjecture counterexample

21 July, 2026 in math.AG | Tags: , | by | 147 comments

The notorious Jacobian conjecture can be formulated concretely over the complex numbers as follows.

Conjecture 1 (Jacobian Conjecture) Let {F:{\bf C}^n \rightarrow {\bf C}^n} be a polynomial map in {n} complex variables, whose Jacobian {\mathrm{det} DF} is a non-zero constant. Then {F} is invertible (with polynomial inverse).

The condition that the Jacobian {\mathrm{det} DF} is non-zero is equivalent to {F} being locally invertible. (The implication of local invertibility from non-vanishing Jacobian follows from the inverse function theorem; the converse implication can be derived from the Weierstrass preparation theorem, but is omitted here; see also Lemma 5 of this previous blog post.) Also, from the fundamental theorem of algebra, once the Jacobian polynomial {\mathrm{det} DF} is non-zero, it must be constant. So the hypothesis “Jacobian {\mathrm{det} DF} is a non-zero constant” can be replaced with “{F} is locally invertible”. So the Jacobian conjecture can be viewed as an assertion that local invertibility implies global invertibility. The complex numbers can be easily replaced with other fields of characteristic zero by the Lefschetz principle, but I prefer to work in the concrete setting of the complex numbers.

It was recently shown (using the Fable AI) that the conjecture is false in three dimensions (and thus in higher dimensions as well):

Theorem 2 (Counterexample to conjecture) There exists a polynomial {F : {\bf C}^3 \rightarrow {\bf C}^3} which has non-zero constant Jacobian, but is not invertible.

The conjecture remains open in two dimensions, and is easy to establish in one dimension.

The example can be stated completely explicitly: one can take

\displaystyle F(z_1,z_2,z_3) = \Big((1+z_1 z_2)^3 z_3 + z_2^2 (1+z_1z_2) (4+3z_1z_2), \ \ \ \ \ (1)

\displaystyle z_2 + 3 z_1 (1+z_1z_2)^2 z_3 + 3 z_1 z_2^2 (4+3z_1z_2),

\displaystyle 2 z_1 - 3 z_1^2 z_2 - z_1^3 z_3\Big)

and one can verify by a brief calculation that

\displaystyle \mathrm{det} DF = -2

and

\displaystyle F(0,0,-1/4) = F(1,-3/2, 13/2) = F(-1,3/2,13/2)

\displaystyle = (-1/4,0,0).

While this is an extremely quick verification, the construction presented in this fashion appears like a massive miracle. The polynomial {F} has degree seven, so a priori the Jacobian {\mathrm{det} DF} ought to be a polynomial in three variables of degree as large as {3 \times 6 = 18}, so the fact that all non-constant coefficients of this polynomial vanish looks like a massive cancellation involving {\binom{18+3}{3}-1 = 1329} equations, which is much larger than the {3 \times \binom{7+3}{3} = 360} degrees of freedom for a generic degree seven polynomial map of three variables. So finding such a polynomial looks highly unlikely to be located by brute force.

The example has since been retroactively explained in more geometric terms. As a “digestion” exercise to myself, I sought to write this explanation with relatively little use of algebraic geometry, in a manner that minimizes the amount of “miracles” required, although there are still a few places where some remarkable phenomena occur.

It is convenient to use the local injectivity formulation, and to generalize the domain {{\bf C}^3} to an equivalent affine variety. Namely, we will show

Theorem 3 (Counterexample, reformulated) There exists an affine variety {X \subset {\bf C}^5} that is isomorphic to {{\bf C}^3} by polynomial changes of variable, and a polynomial map {F : X \rightarrow {\bf C}^3} which is locally injective, but not globally injective.

Clearly one can get from Theorem 3 to Theorem 2 by composing with the isomorphism {X \cong {\bf C}^3} and using the previously mentioned fact that local injectivity implies non-zero constant Jacobian. Our objective is now to find data {X}, {F : X \rightarrow {\bf C}^3} that obeys three separate properties:

  • (a) {F} is locally injective on {X}.
  • (b) {F} is not globally injective on {X}.
  • (c) {X} is isomorphic to {{\bf C}^3} by polynomial changes of variable.
The advantage of splitting the problem in to these three components is that we can build towards each of them separately.

(A pedantic remark: strictly speaking, in the arguments below, we not only replace the domain {{\bf C}^3} of {F} by an equivalent variety {X}, but also replace the range {{\bf C}^3} of {F} by an equivalent variety {V}. But the equivalence between {V} and {{\bf C}^3} is a boring linear isomorphism ({V} will just be a hyperplane in a four-dimensional vector space {\mathrm{Sym}^3({\bf C}^2)}), so we do not highlight this aspect of the construction.)

It turns out that {F} and {X} can be built out of the operation of multiplication of low degree polynomials. Namely, consider the following three simple affine spaces:

  • The space {\mathrm{Sym}^1({\bf C}^2)} of linear homogeneous polynomials {L(z,w) = az + bw} of two complex variables {z,w}.
  • The space {\mathrm{Sym}^2({\bf C}^2)} of quadratic homogeneous polynomials {Q(z,w) = cz^2 + dzw + ew^2} of two complex variables {z,w}.
  • The space {\mathrm{Sym}^3({\bf C}^2)} of cubic homogeneous polynomials {C(z,w) = fz^3 + gz^2w + hzw^2 + iw^3} of two complex variables {z,w}.
(The notation {\mathrm{Sym}^k(V)} here refers to the {k^{th}} symmetric power of a vector space {V}.) Clearly these spaces are isomorphic to {{\bf C}^2, {\bf C}^3, {\bf C}^4} respectively. Furthermore, we have a multiplication map {F : \mathrm{Sym}^1({\bf C}^2) \times \mathrm{Sym}^2({\bf C}^2) \rightarrow \mathrm{Sym}^3({\bf C}^2)}, mapping a pair {(L,Q)} of a linear polynomial {L} and a quadratic polynomial {Q} to a cubic polynomial

\displaystyle F(L,Q) := LQ.

(Right now, the domain and range of this map {F} is larger dimensional than the target of three; we will cut the dimensions down to three as the argument progresses.)

The map {F}, essentially a map from {{\bf C}^5} to {{\bf C}^4}, is clearly polynomial; it is given explicitly in coordinates as

\displaystyle F( (a,b), (c,d,e) ) = (ac, ad + bc, ae + bd, be). \ \ \ \ \ (2)

The map {F} also enjoys two basic (and commuting) symmetries:
  • If one applies a scaling {(L,Q) \mapsto (\lambda_1 L, \lambda_2 Q)} for some non-zero complex numbers {\lambda_1, \lambda_2}, then the product {LQ} is scaled by {C \mapsto \lambda_1 \lambda_2 C}: {F( \lambda_1 L, \lambda_2 Q) = \lambda_1 \lambda_2 F(L,Q)}.
  • If one applies a change of variables {(L, Q) \mapsto (L \circ T, Q \circ T)} for some invertible linear transformation {T \in \mathrm{SL}_2({\bf C})}, then the product {LQ} is transformed by {C \mapsto C \circ T}: {F(L \circ T, Q \circ T) = F(L,Q) \circ T}.
So this map enjoys a huge amount of equivariance, basically with respect to an action of the five-dimensional group {{\bf C}^\times \times {\bf C}^\times \times \mathrm{SL}_2({\bf C})}.

The five-dimensional domain {\mathrm{Sym}^1({\bf C}^2) \times \mathrm{Sym}^2({\bf C}^2)} is of course larger than the four-dimensional range {\mathrm{Sym}^3({\bf C}^2)}, so the map {F} clearly cannot be injective. This can already be seen from the scaling symmetry, as the specific scalings

\displaystyle (L, Q) \mapsto (\lambda L, \lambda^{-1} Q) \ \ \ \ \ (3)

for {\lambda \in {\bf C}^\times} modify the linear and quadratic polynomials {L,Q} but not their product {C = LQ}. But even if one quotients out by this symmetry (3) to cut the dimension of the domain down to four, the map {F} is still not injective for the following basic reason. A generically chosen cubic polynomial {C} will split into the product {C = L_1 L_2 L_3} of three independent linear polynomials. Then there are three pairs

\displaystyle (L_1, L_2 L_3), (L_2, L_1 L_3), (L_3, L_1 L_2) \ \ \ \ \ (4)

 which all map to the same cubic polynomial

\displaystyle F(L_1, L_2 L_3) = F(L_2, L_1 L_3) = F(L_3, L_1 L_2) = C

under the multiplication map {F}, but are not related to each other by scaling symmetry (3). Thus, we see that even after quotienting out by the scaling symmetry (3), the multiplication map {F} is generically non-injective in a three-to-one fashion. Thus we already have achieved something resembling goal (b)!

It will be convenient to “spend” the scaling symmetry {(L, Q) \mapsto (\lambda L, \lambda^{-1} Q)} to obtain a useful normalization. If {L(z,w) = az+bw} is a linear polynomial and {Q(z,w) = cz^2 + dzw + ew^2} is a quadratic polynomial, the (homogeneous) resultant {\mathrm{Res}(L,Q)} can be defined by the determinant

\displaystyle \mathrm{Res}(L,Q) = \begin{vmatrix} a & b & 0 \\ 0 & a & b \\ c & d & e \end{vmatrix} = a^2 e - abd + c b^2. \ \ \ \ \ (5)

If we have a factorization

\displaystyle L(z,w) = a (z - \alpha w), \quad Q(z,w) = c (z - \beta_1 w)(z - \beta_2 w)

then the resultant can also be described as

\displaystyle \mathrm{Res}(L,Q) = a^2 c (\alpha - \beta_1) (\alpha - \beta_2).

Thus the resultant measures whether the linear polynomial {L} and the quadratic polynomial {Q} share a common root. A fundamental fact about resultants is that they are {SL_2}-invariant: for any {T \in SL_2({\bf C})}, we have

\displaystyle \mathrm{Res}(L \circ T, Q \circ T) = \mathrm{Res}(L,Q).

One way to see this is to check it first for translations {(z,w) \mapsto (z + hw, w)} (which translate the roots {\alpha,\beta_1,\beta_2} by {-h} while leaving {a,c} unchanged) and for inversions {(z,w) \mapsto (w,-z)} (which map {\alpha,\beta_1,\beta_2} to {-1/\alpha, -1/\beta_1, -1/\beta_2} while mapping {a,c} to {a\alpha} and {c\beta_1 \beta_2} respectively), and then noting that these transformations generate all of {SL_2({\bf C})}. They also interact very nicely with scaling:

\displaystyle \mathrm{Res}(\lambda_1 L, \lambda_2 Q) = \lambda_1^2 \lambda_2 \mathrm{Res}(L,Q).

In particular, the scaling symmetry (3) multiplies {\mathrm{Res}(L,Q)} by {\lambda}:

\displaystyle \mathrm{Res}(\lambda L, \lambda^{-1} Q) = \lambda \mathrm{Res}(L,Q). \ \ \ \ \ (6)

 Thus, we can (generically) normalize away this scaling symmetry by imposing the condition

\displaystyle \mathrm{Res}(L,Q) = 1. \ \ \ \ \ (7)

We now have a restricted multiplication map (which by abuse of notation we will continue to call {F}) from the four-dimensional variety

\displaystyle \{ (L,Q) \in \mathrm{Sym}^1({\bf C}^2) \times \mathrm{Sym}^2({\bf C}^2) : \mathrm{Res}(L,Q) = 1\} \ \ \ \ \ (8)

to the four-dimensional space {\mathrm{Sym}^3({\bf C}^2)}. This map {F} is still not globally injective, as we can take the three pairs in (4) from before and apply the scaling (3) separately to each of the three pairs to obtain the normalization (7). So we have kept property (b). Furthermore, this map retains the {SL_2}-equivariance (and also one remaining scaling symmetry, though we will not make much further use of that symmetry).

But we now also have property (a)! Suppose we want to show the local injectivity of {F} in the neighborhood of a pair {(L,Q)} with {\mathrm{Res}(L,Q) = 1}. As the resultant is non-vanishing, the root {\alpha} of {L} (which exists in the Riemann sphere, or projective line if you prefer) is distinct from the two roots {\beta_1, \beta_2} of {Q} (though the latter two roots could be equal to each other). Applying the {SL_2} action (which performs Möbius transforms on the roots), one can assume without loss of generality that {\alpha} is the point at infinity (or equivalently {a=0}), thus {L(z,w) = b w} for some complex number {b} and {Q(z,w) = c (z - \beta_1 w)(z - \beta_2 w)} for some complex numbers {c, \beta_1, \beta_2}, with the resultant condition (7) simplifies to {cb^2 = 1} (so in particular {c,b} are also non-zero). It is then clear that if one perturbs {L} and {Q} by a small amount (say, modifying each coefficient by {O(\varepsilon)}), then the root {\alpha=\infty} of {L} will perturb to something large ({\gg 1/\varepsilon}), while the roots {\beta_1,\beta_2} of {Q} stay bounded. Thus, just from knowledge of the product {F(L,Q)}, one can reconstruct which of the three roots of this cubic polynomial will be the perturbed root of {L}, and which two will be the perturbed roots of {Q}; from this and (6), (7) we can also reconstruct the leading coefficient {c} of {Q}, and this completely determines both {L} and {Q}. This establishes the local injectivity property (a). (In fact it is étale, but we will not need the machinery of étale maps here.)

Unfortunately, (the four-dimensional analogue of) condition (c) fails: the quadric hypersurface (8) is not isomorphic to the affine space {{\bf C}^4}. But we can try to get around this by passing to a three-dimensional slice. Let {V} be some three-dimensional affine plane of {\mathrm{Sym}^3({\bf C}^2)} (which we will take to avoid the origin for technical reasons), then we can restrict {F} as a map from the set

\displaystyle \{ (L,Q) \in \mathrm{Sym}^1({\bf C}^2) \times \mathrm{Sym}^2({\bf C}^2) : \mathrm{Res}(L,Q) = 1; \ \ \ \ \ (9)

\displaystyle F(L,Q) \in V\}

to {V}. {V} is clearly identifiable (by linear changes of coordinate) to {{\bf C}^3}. As {F} was already locally invertible, it remains locally invertible under restriction; and because generic cubic polynomials {C} had three preimages under {F} in (8), this continues to be the case after restricting to (9) (unless {V} was somehow so degenerate that it had no generic elements, but this turns out to be impossible). So we have retained properties (a) and (b). The miracle is that, with a good choice of {V}, we can also obtain (c) and obtain the desired counterexample to the Jacobian conjecture: despite appearances, the variety (9) is in fact equivalent to the affine space {{\bf C}^3} by polynomial changes of variable!

Let’s see how. The affine hyperplanes in {\mathrm{Sym}^3({\bf C}^2)} avoiding the origin are parameterized by the dual space of {\mathrm{Sym}^3({\bf C}^2)} avoiding the origin, which one can think of as the non-zero third order homogeneous differential operators {D = j \partial_z^3 + k \partial_z^2 \partial_w + l \partial_z \partial_w^2 + m \partial_w^3} in two variables. Indeed, every such operator {D} generates an affine hyperplane {\{ C \in \mathrm{Sym}^3({\bf C}^2) : D(C) = 1\}} that avoids the origin, and conversely by duality every affine hyperplane avoiding the origin arises in this form uniquely. Just as the cubic polynomials in {\mathrm{Sym}^3({\bf C}^2)} can be factored into three linear polynomials, the differential operators in the dual space {\mathrm{Sym}^3({\bf C}^2)^*} can also be factored into three linear differential operators, e.g.,

\displaystyle D = j (\partial_z - \gamma_1 \partial_w) (\partial_z - \gamma_2 \partial_w) (\partial_z - \gamma_3 \partial_w)

in the case that {j} is non-zero. The {SL_2} action moves the roots {\gamma_1,\gamma_2,\gamma_3} around the Riemann sphere by Möbius transformations. As these transformations are {3}-transitive, the actual selection of such roots is not too important (and the scaling symmetry similarly makes the choice of leading coefficient {j} unimportant); the only thing to keep track of is whether the roots repeat. Up to the symmetries, there are in fact just three different equivalence classes of differential operator {D} (and thus of affine hyperplane {V}) to consider:
  • Operators where the three roots {\gamma_1,\gamma_2,\gamma_3} are all distinct, thus {D = D_1 D_2 D_3} for independent first-order operators {D_1,D_2,D_3}.
  • Operators where two roots coincide and one is distinct, thus {D = D_1^2 D_2} for independent first-order operators {D_1,D_2}.
  • Operators where all three roots coincide, thus {D = D_1^3} for some first-order operator {D_1}.

It turns out that the affine miracle for (9) occurs precisely in the second case, when {D} has two identical roots. I do not have a completely satisfactory geometric explanation for this miracle, but one can verify it by the following coordinate computation.

By applying the {SL_2} action, we can normalize so that {D = \frac{1}{2} \partial_z^2 \partial_w}, thus {V} is now the affine hyperplane of cubic polynomials {C(z,w) = f z^3 + g z^2 w + h z w^2 + i w^3} with {g=1}. Using (2) and (5), the variety (9) can now be described explicitly in coordinates as

\displaystyle \{ (a,b,c,d,e) \in {\bf C}^5 : a^2 e - abd + cb^2 = 1; ad + bc = 1 \}. \ \ \ \ \ (10)

At first glance this seems to be a generic-looking variety cut out by a cubic equation and a quadratic equation – hardly a candidate to be affine! But observe that if {a} is non-zero, then the second equation {ad+bc = 1} can be solved for {d},

\displaystyle d = \frac{1 - bc}{a} \ \ \ \ \ (11)

 and the first equation {a^2 e - abd + cb^2 = 1} can be solved for {e},

\displaystyle e = \frac{1 + abd - cb^2}{a^2}. \ \ \ \ \ (12)

 Putting these two equations together, we see that as long as one removes the case {a=0}, the quintuple {(a,b,c,d,e)} is uniquely determined by {(a,b,c)} by a change of variables which is Laurent in {a} and polynomial in {b,c}. Thus we have a nice birational equivalence

\displaystyle \{ (a,b,c,d,e) \in {\bf C}^5 : a^2 e - abd + cb^2 = 1; ad + bc = 1; a \neq 0 \}

\displaystyle \cong \{ (a,b,c) \in {\bf C}^3 : a \neq 0 \}.

Thus we have already almost established property (c): the variety (9) becomes birationally equivalent to {{\bf C}^3} after cutting out the {a=0} subvariety. In particular, for each fixed non-zero value {a_0} of {a}, the corresponding fiber

\displaystyle \{ (a,b,c,d,e) \in {\bf C}^5 : a^2 e - abd + cb^2 = 1; ad + bc = 1; a = a_0 \}

of (10) is equivalent to {{\bf C}^2} by polynomial changes of variable, since we can reconstruct {d,e} from the coordinates {b,c} by the polynomial formulae

\displaystyle d = \frac{1-bc}{a_0}; \quad e = \frac{1 + a_0 d b - c b^2}{a_0^2}.

So we just need to glue back in the {a=0} fiber. Indeed, from (10) we see that the fiber at {0} is just

\displaystyle \{ (0,b,c,d,e) \in {\bf C}^5 : cb^2 = 1; bc = 1 \}.

Now we observe a key miracle: the cubic equation {cb^2 = 1} and quadratic equation {bc = 1} have a unique affine solution {b=c=1} (as opposed to the six possible solutions that Bezout’s theorem might suggest – the other five solutions live on the line at infinity). So the fiber here is also affine:

\displaystyle \{ (0,1,1,d,e) \in {\bf C}^5 : d, e \in {\bf C} \}.

This is extremely encouraging for the purposes of establishing property (c), as it strongly suggests that the variety (10) has the structure of an {{\bf C}^2}-bundle over {{\bf C}^1}, which is already extremely close to being isomorphic to the affine space {{\bf C}^3}. The main remaining task is to make sure that nothing singular happens in the limit {a \rightarrow 0}, and that a global polynomial coordinate chart for (10) that covers both the {a \neq 0} and {a = 0} fibers can be constructed.

The standard way to proceed here is to manipulate various tangent spaces using the modern machinery of algebraic geometry and commutative algebra, but given my own background, I prefer to adopt the language of analysis, and in particular big-O notation (in place of the ideals used in algebraic geometry), in order to investigate the limit {a \rightarrow 0} by hand. On the variety (10), let us use {O(X)} to denote any multiple of {X} by a polynomial expression in {a,b,c,d,e}. Thus, for instance, the equation {ad + bc = 1} implies that

\displaystyle bc = 1 + O(a) \ \ \ \ \ (13)

while the equation {a^2 e - abd + cb^2 = 1} implies that

\displaystyle cb^2 = 1 + O(a) \ \ \ \ \ (14)

 as well as the more refined estimate

\displaystyle cb^2 = 1 + abd + O(a^2). \ \ \ \ \ (15)

 In the {a=0} case we could conclude that {b=c=1}. Now we perturb this observation. Multiplying (13) by {b} we have {b^2 c = b + O(a)}, which on substitution into (14) gives {b = 1 + O(a)}; substituting this back into either (13) or (14) also gives {c = 1 + O(a)}.

We can get some more precise asymptotics by also taking advantage of (15). Substituting {ad+bc=1} into (15), we obtain after some algebra

\displaystyle 2 cb^2 = 1 + b + O(a^2).

So if we write {b = 1+O(a)} more explicitly as {b = 1 + a y}, then we have

\displaystyle 2 c (1 + 2ay + O(a^2)) = 2 + ay + O(a^2)

and thus

\displaystyle c = 1 - \frac{3}{2} ay + O(a^2). \ \ \ \ \ (16)

 Substituting this back into (11) gives an asymptotic for {d}:

\displaystyle d = \frac{1 - bc}{a}

\displaystyle = \frac{1 - (1 + ay) (1 - \frac{3}{2} ay + O(a^2))}{a}

\displaystyle =\frac{1}{2} y + O(a).

Finally, one can insert these estimates into (12), although one only gets a trivial bound in this case:

\displaystyle e = \frac{1 + abd - cb^2}{a^2}

\displaystyle = \frac{1 + a (1+O(a)) (\frac{1}{2} y + O(a)) - (1 - \frac{3}{2} ay + O(a^2)) (1 + ay)^2}{a^2}

\displaystyle = O(1).

Expanding the {O(a^2)} error term in (16) as {a^2 z}, and doing a little more algebra, we thus have a polynomial change of variables

\displaystyle a = a

\displaystyle b = 1 + ay

\displaystyle c = 1 - \frac{3}{2} ay + a^2 z

\displaystyle d = \frac{1-bc}{a} = \frac{1}{2} y - az + \frac{3}{2} a y^2 - a^2 yz

\displaystyle e = \frac{1 + abd - cb^2}{a^2} = -2z + 4y^2 - 4ayz + 3ay^3 - 2a^2 y^2 z

which completely parameterizes the variety (10) by polynomial combinations of three coordinates {a,y,z}. This already gives (c) and thus completes the proof of Theorem 3.

The previous computations, when expanded out, also gives polynomial inverse maps:

\displaystyle a = a

\displaystyle y = 2bd - ae

\displaystyle z = 2d^2 + ce + 6bd^2 + 3bce - \frac{9}{2} e

The map from {(a,y,z)} to the {(f,h,i)} coefficients of {F(L,Q)} (dropping the {g} coefficient which is constrained to equal {1}), we obtain a polynomial map

\displaystyle (a,y,z) \mapsto (G_1(a,y,z), G_2(a,y,z), G_3(a,y,z))

with

\displaystyle \begin{array}{rl} G_1(a,y,z) &= a - \frac{3}{2} a^2 y + a^3 z \\ G_2(a,y,z) &= \frac{1}{2} y - 3az + 6ay^2 - 6a^2 yz + \frac{9}{2} a^2 y^3 - 3a^3 y^2 z \\ G_3(a,y,z) &= -2z + 4y^2 - 6ayz + 7ay^3 - 6a^2 y^2 z + 3a^2 y^4 \\ & \quad - 2 a^3 y^3 z \end{array}

which theory predicts to have a constant Jacobian, and indeed one can calculate that the Jacobian is {-1}. This is essentially the original example up to trivial changes of variable; indeed, one can check that the map

\displaystyle (a, y, -2z) \mapsto (G_3(a,y,z), 2G_2(a,y,z), 2G_1(a,y,z))

is exactly the map {F} given in (1).

AI disclosure: I used an AI chatbot to discuss various aspects of this problem and to confirm several of the calculations made here.

Two more apps: visualizing the zeta process and the motions of the heavens

16 July, 2026 in expository, math.NT, math.PR | Tags: , , , , | by | 14 comments

I believe that the creation of visualization apps to illustrate mathematical or scientific concepts is a particularly favorable use case for modern coding agents, as many of the downside risks attached to other LLM use cases are limited:

  1. Not mission-critical. As such apps are not authorative sources of truth and only used for secondary purposes, a small positive error rate in the output can be acceptable.
  2. Stand-alone. As the applets are not destined to be incorporated into a larger codebase or literature, the technical debt incurred by delegating all the coding to an LLM agent is bounded.
  3. End product is deterministic (and sandboxed). As the applets run on a deterministic language (Javascript), are sandboxed against file or internet access, and do not make any LLM calls at run-time, security and privacy concerns are minimal, and the applet can be maintained without continued premium LLM access or resource-intensive compute.
  4. Not replacing primary skills. While deskilling is the tradeoff one accepts when relying on these tools to accelerate output, I am perfectly willing to forego the opportunity to keep my Javascript skills at a high level, as this is a tertiary skill for me at best in my chosen profession. (I continue to manually program in Lean and in Python to keep in practice with programming in general.)
  5. Not competing with humans. To my knowledge, there is no existing human effort that is being duplicated by these applets (the activity in this direction appears to have peaked two decades ago).

I would however caution against unrestricted LLM use when one or more of the above five favorable situations is not in effect.

With these points in mind, I have used such an agent to create two further apps. The first app illustrates the “zeta process” that was introduced in my recent paper with Alexeev, Barreto, Li, Lichtman, Price, Shah, and Tang, though it was first discovered by an AI. For each s > 1, the zeta distribution Z_s is a random natural number with distribution

\displaystyle {\bf P}( Z_s = k ) = \frac{1}{\zeta(s) k^s}.

It has long been known that this distribution has good number-theoretic properties: for instance, the number of times a given prime p divides Z_s has a geometric distribution of mean p^{-s}. However, the new observation is that these random variables Z_s can be chained together into a single stochastic process, which we call the “zeta process”, which is an infinite divisibility chain. I used an agent to create an app to visualize this process:

BERJAYA

The underlying process is generated by several exponential random variables at each prime: in the above instantiation of the process, two such variables are visible at the prime p=2, and one variable at the primes p=3,5. At a given choice of s, Z_s is formed by collecting all the variables below this threshold (and for which all predecessors also lie below the threshold); in the above illustration, this amounts to one variable at each of the primes p=2,3,5, leading to Z_2 = 30 in this case. Additional visualizations in the app display the distribution of each Z_s, as well as the distribution of the hitting probability \nu_\Lambda, which among other things can be used to give a quick solution to Erdős problem #1196.

The second app is rather different in nature, and is a somewhat whimsical attempt to display the motion of the heavens, both at “human” scales of space and time, and at more “astronomical” scales (in which the motion of the planets in particular are more apparent). It is very loosely inspired by the game “Katamari Damacy“, in which one absorbs both terrestrial and celestial objects of many different scales. Here is how the app typically looks at a human scale:

BERJAYA

And here is how it looks when one’s perspective leaves the Earth’s atmosphere:

BERJAYA

(As I did not want to render an entire explorable world in this app, the observer in the app is only limited to changing his or her size, from a human to a creature of comparable size to the Earth itself; they cannot move horizontally on the planet.) At the largest scales of space and time, the classic orrery diagram appears:

BERJAYA

After lengthy conversations with the agent, I was able to implement many astronomical phenomena, including phases of the Moon, the effect of Earth’s rotation against the fixed stars (though one can also stabilize one’s view against those stars to see the Earth’s rotation more directly), and so forth.

Visualizing the Gilbreath expectation sequence

14 July, 2026 in expository, math.PR | Tags: , | by | 6 comments

One byproduct of learning how to use coding agents to create visualization apps is that it now becomes straightforward to convert any figure in one’s papers that had already been generated by code (e.g., in Python) into a more interactive, animated applet.

I can illustrate this with Figure 1 from my recent paper on the Gilbreath conjecture with Chase and Hunter, reproduced below:

BERJAYA

This plot displays both exact and numerically simulated values of a certain poorly understood sequence c_n relating to the Gilbreath conjecture, which I will call the “Gilbreath expectation sequence” here for lack of a better name. The definition of the sequence is as follows. Consider a “Gilbreath array” which is an inverted pyramid, where the top entries are n+1 independent exponential random variables of mean 1, and all the other entries are the absolute values of the differences of the two entries immediately above it. Thanks to the visualizer app, I can quickly give an example (with n+1=6):

BERJAYA

The left diagonal entries are then random variables; the sequence c_0,\dots,c_n are defined to be the expectation of these values. (The process is stationary, so in fact any entry on the i^{th} row will have expectation c_i.)

If one starts with the first n normalized prime gaps (which have expectation about \log n/2, and are conjecturally distributed asymptotically according to a geometric distribution), then standard conjectures (e.g., the prime tuples conjecture) predict that the k^{th} row entries should decay like c_k\log n/2, at least for small k. So the Gilbreath conjecture appears to be tied to how fast the sequence c_k decays with k.

One can in principle work out each value of c_k as an explicit rational number by performing a certain complicated multivariate integral, but in the paper we only did this for k \leq 3 (the orange line in the above figure); for the remaining k we performed a Monte Carlo simulation with 10^6 Gilbreath arrays to obtain a numerical approximation (in blue), which (as per the law of large numbers) agreed well with the theoretical values. A later calculation of Michael Ross extended the theoretical values to k \leq 6, maintaining the good fit:

BERJAYA

The asymptotic behavior of the sequence c_n remains mysterious. Clearly, it is not monotonic; in fact we cannot even prove it is bounded. The best we could do in our paper was establish an inequality which, roughly speaking, showed that c_n cannot decay faster than 1/n.

In a recent preprint of Ross, these numerics were extended, and a rough empirical prediction

\displaystyle c_n \approx C \lambda^{s_2(n)} / n

was proposed for some constants C>0 and \lambda>1 (empirically \lambda \approx 1.17), where s_2(n) is the number of 1’s in the binary expansion of n; in particular, it is the fluctuation in this quantity s_2 that is intended to explain much of the non-monotonic behavior of c_n. These are now all displayed in the following companion applet, which was a routine matter to generate in about an hour by the coding agent (which by this point has extensive experience with creating such apps, encoded via a “skill” markdown file that it maintains):

BERJAYA

The appearance of the quantity s_2(n) may initially appear mysterious, but it is related to Lucas’s theorem, Kummer’s theorem, and the Sierpinski gasket. Consider for instance a Gilbreath array where all the entries are zero except for a single “spike”. Then the following Sierpinski pattern emerges:

BERJAYA

Here is what an n=64 version of this picture looks like (with the spike positioned at the 32th entry):

BERJAYA

The number of 1s in the k^{th} row is then 2^{s_2(k)} (if we index the rows starting from zero), which is at least of the same shape as the empirical prediction, albeit with different constants. (This sequence is also known as Gould’s sequence.)

Numerically, we seem to observe fragments of Sierpinski gaskets being generated before decaying (often due to “collisions” with other gaskets):

BERJAYA

However, it is not clear to me at all what the asymptotic probabilistic model should be, even heuristically; it does not resemble any random shape model that I am familiar with. But perhaps there are readers more expert in probability theory or statistical physics who may be able to suggest such an asymptotic limit?

Call for long programs, workshops, and summer schools at IPAM

14 July, 2026 in advertising | Tags: | by | 1 comment

(I am writing here in my capacity as Director of Special Projects at IPAM.)

IPAM seeks program proposals from the mathematical, statistical, and scientific communities for long programs, workshops, and summer schools.  Most program proposals are reviewed at IPAM’s Science Advisory Board meeting, held in November each year.  Programs are selected on the basis of their scientific impact and contribution to IPAM’s goals. IPAM is committed to supporting a community where people of all backgrounds and points of view can engage, learn, and thrive.  If you would like to discuss your program ideas and prepare a proposal for IPAM’s consideration, you are encouraged to contact the IPAM Director. For more information visit: https://www.ipam.ucla.edu/propose-a-program/long-programs-2/

A paper diagram visualizer

13 July, 2026 in advertising | Tags: | by | 16 comments

I am finding the newly revealed capability to code old applet ideas into reality to be very tempting to sink more time into, though I am certainly encountering the common “vibe coding” experience that the process can produce something that superficially resembles a finished product well before a satisfactory level of testing and review has been completed; indeed, it is the review process which is now the most time-consuming, to the point where I think any further advances in coding agent capability will have little impact on the new bottlenecks in the design process.

In any event, I spent a few hours working to realize a proposal I had made back in 2023 to automatically create diagrams to visually illustrate the logical flow of a given mathematical paper. At the time, Freddie Manners, extrapolating from the half-decent capability of the then-newly released ChatGPT 3.5 at this task, presciently predicted that “by the time a dedicated tool had been completed, the next general purpose engine would be better than it”.

With that in mind, I decided to focus not on the generation of the diagram – which now can be done at various levels of quality by any number of large language models – but on its presentation. The result is the following app, which can take a certain formatted JSON file of dependencies between theorem objects and produce an interactive graph which can be explored, edited and also exported (somewhat lossily) into other standard formats such as SVG, TikZ, quiver, or Mermaid. Here is a screenshot of a diagramming of the celebrated proof by Wang and Zahl of the three-dimensional Kakeya conjecture:

BERJAYA

Using an LLM, I generated diagrams for eight papers for demonstration purposes, including for instance a diagram for Wiles’s proof of Fermat’s last theorem, or of Szemeredi’s proof of his famous theorem on arithmetic progressions (which sports a notoriously convoluted such diagram in the original paper), as well as a few papers of my own. If there are other requests to diagram particular papers, I can try to use an LLM to generate more examples; but my intention is for users of the app to create their own such diagrams, either by manually constructing them, or by directing their own AI tools to build the diagram in the required format (which is a JSON, with the precise specification given here).

I mentioned in the previous post that for these sorts of visualization apps, which work deterministically for a given set of inputs, the downside risk of LLM use to build the app is acceptably low. For this particular app, there is a complicating factor, which is that while the app does remain deterministic, the data I used to populate the app – namely, the above diagrams – are also LLM-generated. I have done spot checks comparing the diagrams against the source papers, and did not find any errors; however, they are not guaranteed to be 100% accurate, and should only be used as approximations to the logical structure of these papers rather than completely exact representations. (The latter might become deterministically extractable should the results of these papers become formalized in a proof assistant language, but this is not currently the case.) Still, I hope these sorts of diagrams can serve as a helpful initial guide when first trying to read and understand a complex paper.

A random variable visualizer

12 July, 2026 in advertising, math.PR | Tags: , , | by | 7 comments

With the advent of modern coding agents, many visualization projects that I had proposed in the past, but dropped due to the time and complexity of the coding portion of the task, have now become relatively feasible, in that a reasonable quality prototype (suitable for non-mission-critical tasks such as providing secondary visual aids, where it is not absolutely necessary that the product is 100% bug-free) can now be generated in a matter of hours using such tools. I don’t immediately plan to work on my entire backlog of such projects, but I did spend a few hours this weekend on one such project, namely the proposal from this 2016 blog post to visualize random variables as animated quantities, which can be either viewed numerically or displayed as a scatterplot. After some back and forth with the coding agent, I was able to come up with a working app. Here is a screenshot of the app displaying a visualization of Berkson’s paradox, which asserts that independent variables can become correlated to each other after applying a conditioning:

BERJAYA

The app in fact is a “compiler” for a small, custom programming language (think of a simplified hybrid of Python and Excel) which allows for the introduction of random variables, manipulates them through operations such as arithmetic operations or conditioning, and then plots them as animations or as text. (The screenshot above is static, but when the app is live, it will update at the indicated speed.)

As always, I would be happy to receive feedback on the app, which I hope can be useful as a visual aid to understand basic probabilistic concepts such as independence or conditioning.

Old and new apps, via modern coding agents

11 July, 2026 in advertising | Tags: , , , | by | 10 comments

I have been interested in machine-assisted ways to do and teach mathematics from as far back as 1999, when I started coding several applets in Java 1.0, both for my complex analysis and linear algebra courses, as well as to visualize various mathematical objects I was interested in (such as honeycombs or Besicovitch sets). This was moderately successful; but the applets were time-consuming to program. Eventually, the standards for web pages stopped supporting this version of Java, and the applets became non-functional.

However, in the last few days I have begun the process of migrating much of my old web page and blog data to a more maintainable repository, using modern AI assistance. As an experiment, I asked the agent to port my old applets to a modern supported language (we landed on Javascript), and it managed to do so in a matter of hours, with all of my old applets now functional again, with even a few graphical upgrades (for instance, the Besicovitch set applet is now colorized, in contrast to my original monochrome version). I am particularly pleased to see the honeycomb applet that I wrote with Allen Knutson in 1999 come back to life, as this was a particularly tricky one to code by hand:

BERJAYA

Notoriously, LLM-based coding agents can create various blatant or subtle bugs in their code; but in the porting of these two dozen or so applets, I could only find one minor bug (the handling of a drag event in one of the complex analysis applets had unwanted behavior when dragging outside of the main box), and in fact the agent identified two bugs in the original code that I was not aware of, so it ended up being a net wash as far as code quality was concerned. In any event, as these applets are meant to be secondary visual aids rather than critical components of a mathematical argument, the downside risk of such bugs is relatively low.

The process was painless enough that I decided to also try coding some new apps, in addition to porting the old ones. Back in 1999 I had an ambitious idea for a visualization tool for special relativity; this was before the release of the software tool Inkscape, but the idea I had in mind was basically “Inkscape, but in Minkowski space”. I had even started writing Java code for this app, but the code complexity became too much for me, and I abandoned the project. However, after a couple hours of “vibe coding” with an AI agent, I was finally able to generate an applet that matched the vision I had back in 1999, which can now be found here. A summary of the conversation I had with the agent to generate this code can be found here (it has been edited down to remove a large number of tedious technical implementation reports). While I have playtested the app somewhat, I would be interested in receiving further feedback on this “alpha” version of the applet, as I am sure (especially given the LLM-generated nature of the code) that there are still some bugs and rough edges to be ironed out.

BERJAYA

After writing my blog post on the Gilbreath conjecture paper earlier today, I realized that I could similarly ask the agent to code a visualization tool for the Gilbreath conjecture to accompany the paper and blog post. After another few hours of conversation, this is now done; you can try out the visualization here. Again, the procedure was quite painless (see this transcript of the process), and I think I may add such interactive visualizations as supplements for future papers; as such supplements are not mission-critical to the core of the paper, I again feel that the downside risk of using guided interaction with LLM agents to generate such visualizations is acceptable.

BERJAYA

Gilbreath’s conjecture: a Cramér random model and a deterministic analysis

11 July, 2026 in math.CO, math.NT, math.PR | Tags: , , , | by | 9 comments

Zachary Chase, Zach Hunter and I have uploaded to the arXiv our preprint Gilbreath’s conjecture: a Cramér random model and a deterministic analysis. This paper is motivated by a notorious conjecture of Gilbreath (also proposed eighty years prior by Proth), which one can state as follows: if one starts with the sequence of primes and repeatedly takes absolute differences of consecutive terms, then the first term of each subsequent row is always {1}:

\displaystyle \begin{array}{ccccccccccc} 2 & & 3 & & 5 & & 7 & & 11 & & 13 \\ & 1 & & 2 & & 2 & & 4 & & 2 & \\ & & 1 & & 0 & & 2 & & 2 & \\ & & & 1 & & 2 & & 0 & & \\ & & & & 1 & & 2 & & & \\ & & & & & 1 & & & & \end{array}.

Coming from a PDE background, I like to think of this conjecture as a (discrete) nonlinear “wave equation” problem, where the primes are the “initial data”, the downward direction in the above pyramid is the arrow of “time”, and the “equation of motion” is that the value of the “scalar field” at any given point in “spacetime” is the absolute difference of the values of the two points directly above it. We will informally refer to solutions to such an “equation” as “Gilbreath arrays”.

Numerically, the conjecture has been verified for the first {3.34 \times 10^{11}} rows by Odlyzko. Asymptotically, the conjecture can be heuristically justified as follows. Firstly, because all primes other than {2} are odd, it is easy to see that the first term of each row is odd, while all other terms are even. Next, if one starts with the first {n} primes for some large {n} and takes initial differences, then the prime number theorem tells us that the average size of the next row is about {\log n}, and Cramér’s conjecture predicts that the maximum size should be {O(\log^2 n)}. With each new row, the maximum size can only decrease (since {|a-b| \leq \max(a, b)} for any natural numbers {a,b}), and so one would expect it likely on each row that the maximum size should drop by at least {1} (unless it has already reached {2}). Since there are {n-1} rows to go before one reaches the end, it seems extremely likely that the maximum size should drop down to at most {2} by then, at which point the result is forced from parity reasons.

However, it seems well beyond current technology to try to make these heuristics rigorous; even the first step of proving Cramér’s conjecture is far out of reach. In our paper, we consider two more feasible directions:

  • What is a realistic probabilistic model of the primes, and can one confirm the (asymptotic version of the) conjecture almost surely for such a model?
  • Can one use deterministic arguments to reduce the (asymptotic) Gilbreath conjecture to more tractable looking (and heuristically plausible) statements about iterated differences of primes?

Let us first discuss the question of analyzing probabilistic models. One can strip away the first row and initialize using prime gaps {p_{n+1}-p_n} rather than primes; it is convenient to also strip away the aforementioned parity structure, by eliminating the initial gap {3-2=1}, and dividing all remaining gaps by {2}, so that one now works with an initial sequence {\frac{p_{n+1}-p_n}{2}, n \geq 2} with no parity bias. The conjecture is now equivalent to the first row always being {\{0,1\}}-valued:

\displaystyle \begin{array}{ccccccccccccccccc} 0 & & 0 & & 1 & & 0 & & 1 & & 0 && 1 && 2 && 0\\ & 0 & & 1 & & 1 & & 1 & & 1 && 1 && 1 && 2\\ & & 1 & & 0 & & 0 & & 0 && 0 && 0 && 1\\ & & & 1 & & 0 & & 0 & & 0 && 0 && 1\\ & & & & 1 & & 0 & & 0 && 0 && 1\\ & & & & & 1 & & 0 & & 0 && 1\\ & & & & & & 1 & & 0 && 1\\ & & & & & & & 1 && 1 \\ & & & & & & & & 0 \end{array}.

The Cramér model suggests that the first {n} normalized prime gaps should behave like geometric random variables of mean about {\frac{\log n}{2}}. My co-author, Zachary Chase, established an analogue of the Gilbreath conjecture for a more slowly growing model. Here is a special case of his main theorem:

Theorem 1 Suppose the initial row entries {a_n} of a Gilbreath array are drawn independently from a uniform distribution on {\{0,\dots,f(n)-1\}} for some {2 \leq f(n) \leq \frac{1}{10} \frac{\log\log n}{\log \log \log n}}. Then almost surely, all but finitely many of the rows have a {\{0,1\}}-valued first entry.

The Cramér model morally corresponds to a value of {f(n)} comparable to {\log n}, which is too large for the above theorem to apply. However, we were able to improve the argument, basically allowing {f(n)} to be anything of size {o(n)}. Furthermore, it was not necessary that the distribution be uniform: the important hypothesis was that the distribution not be concentrated in any {2}-separated set, such as the even numbers, the odd numbers, or the multiples of {3}. (See the paper for the precise formulation of “non-concentrated”.) Such a hypothesis is needed since if for instance all initial entries were divisible by {3}, then this property would propagate down the array, and it would become extremely unlikely that the initial values would remain {\{0,1\}}-valued. Our hypotheses are obeyed by the Cramér random model, and so we obtain a heuristic confirmation of the original Gilbreath conjecture for the primes.

One can informally explain our proof of the above result as follows. We consider the portion of the array generated by the first {n} values {a_1,\dots,a_n} for some large {n}. Suppose that at some point {P} deep in this portion of the array, a value {d} that is larger than {1} is attained. Then the two values {a,b} above {P} must satisfy the equation {|a-b|=d}. So, either one of these values is at least {d+1}, or one of them is {d} and the other is {0}. If one iterates this observation, one sees that {P} is the base of an upside-down triangle of {\{0,d\}} values, topped off by at least one location {P'} where the value is at least {d+1}. If one iterates that observation in turn, we see that {P} forms the base of a “tower” of upside-down triangles stacked atop each other, with the number of such triangles bounded by the maximum size of the initial data (in the “backwards light cone” of {P}). In the {f(n)=o(n)} regime, it turns out that the number (or “entropy”) of such towers is subexponential in {n}. So if we can show that each tower only can be created with an exponentially small probability, we can conclude by the standard techniques of the union bound and the Borel–Cantelli lemma.

At this point we use the following elementary observation. Suppose that some finite Gilbreath array coming from say initial data {a_1,\dots,a_n} has been generated, and consider the effect of adding a new value {a_{n+1}} to the initial data, which then triggers {n} iterations of the absolute value difference operation {a \mapsto |a-c|} for various values of {c} until one reaches the new bottom vertex of the array. This difference operation {a \mapsto |a-c|} has the property that the preimage of any {2}-separated set is still {2}-separated. Iterating this, we see that the set of values that make {a_{n+1}} iterate to a {\{0,d\}}-valued bottom vertex is also {2}-separated. So as long as the distribution of {a_{n+1}} avoids {2}-separated sets, one can iterate this observation in {n} to show that it is exponentially rare that large triangles of {\{0,d\}}-valued vertices can be created.

We also consider an asymptotic continuous random model, in which the initial data {a_n} are not natural numbers, but instead independently random non-negative real numbers with an exponential distribution, which we can normalize to have mean {1}; this heuristically is an approximate model for the Gilbreath array generated by the first {n} normalized prime gaps, after dividing by the mean {\log n/2}. In this normalized model, each entry of the {k^{th}} row ends up having the same mean {c_k}. The first few values of {c_k} can be computed explicitly

\displaystyle c_0 = 1; c_1 = 1; c_2 = \frac{7}{9}; c_3 = \frac{227}{288}.

However, the asymptotic behavior of {c_n} remains unclear to us. We were able to show an inequality {c_0 + \dots + c_n \geq \log(n+e)} for any {n}, indicating that {c_k} cannot decay faster than {1/k}, but we do not know whether this is the true decay rate. In any case a decay rate of {1/k} (which is very weakly supported by numerical evidence) is consistent with the Gilbreath conjecture, as it would indicate that the Gilbreath array from the first {n} prime gaps should end up being almost entirely {\{0,1\}}-valued by merely {O(\log n)} steps, well before the {n} steps needed to reach the bottom of the array.

Now we turn to deterministic analysis of Gilbreath arrays. Suppose we found some initial data {a_1,\dots,a_N} that did not grow too quickly (e.g., one had a Cramér-type bound {a_N = O(\log^2 N)}), but still iterated to a final value that was not {\{0,1\}}. What features of the initial data could generate such a failure of a Gilbreath-type conjecture? One way in which the conjecture could fail is if the Gilbreath iteration somehow produced a reasonably long consecutive string of zeroes (say, longer than {\log^{10} N}), as then the next few iterations would not act to decrease the magnitude of the non-zero entries bordering this string of zeroes. Such a scenario would be heuristically rate, as the parity of each element of the array can be worked out explicitly using the parity identity {|a-b| = a + b \hbox{ mod } 2}, and so constant-parity sequences of length say {\log^{10} N} should be almost surely non-existent asymptotically by standard probabilistic heuristics.

Another bad scenario is if the Gilbreath iteration, after some medium number {i} of iterations, produced an extremely long consecutive block (say of length {\exp(\log^{1/10} N) i}) which was entirely {\{0,d\}}-valued for some {d \geq 2}. This block would then persist as a {\{0,d\}}-block for a large number of iterations (equal to the length of the block), thus potentially delaying for a significant time the drop-down of the maximal value to below {d}. For odd {d}, one can use the parity analysis alluded to earlier to argue that the formation of such a block is extremely unlikely; but for even {d}, we can only use such heuristics if we make strong assumptions of joint independence, as we did in the probabilistic analysis in our paper.

In any event, we were able to use purely elementary methods to establish an “inverse theorem” that states, roughly speaking, that the above two scenarios are the only ways in which a Gilbreath array can fail to have a {\{0,1\}}-valued first entry. This basically arises from a more careful analysis of the towers of triangles alluded to earlier. (A previous argument involved considering ways to pack a large triangle by smaller triangles, leading to a MathOverflow question which was nicely answered by Fedja Nazarov and Anders Martinsson, but we later managed to optimize the argument to the point where the answer to this packing question was no longer needed.) So this in principle reduces the (deterministic) Gilbreath conjecture to several more tractable-looking (though complicated to state) assertions, though proving those latter statements seems well out of reach at the moment.

A digestion of unit distance constructions

3 July, 2026 in expository, math.CO, math.NT | Tags: , , | by | 14 comments

Suppose that one has a set {P} of {n} points in the plane, which we will think of as the complex plane {{\bf C}}. Let {d_1(P)} denote the number of unit distances determined by these points, i.e., pairs of points {p,q \in P} whose displacement {w = q-p} obeys the equation

\displaystyle w \overline{w} = 1. \ \ \ \ \ (1)

(It makes little difference for the asymptotics, but we will count the pair {(q,p)} separately from {(p,q)} here.)

The Erdös unit distance problem asks, for a given large number {n}, what is the largest possible value of {d_1(P)} amongst all sets {P} of cardinality {n}?

For instance, if one takes {P} to be {n} equally spaced collinear points with unit spacing, one can obtain a linear construction with {d_1(P) = 2n-2}. Erdös observed that one can improve this construction asymptotically:

Theorem 1 (Erdös construction) There exists point sets {P} of arbitrarily large cardinality {n} such that {d_1(P) \gg n^{1 +c / \log\log n}} for some absolute constant {c > 0}.

In fact, in the construction one could take {c} arbitrarily close to {\log 2}. Erdös famously asked whether {d_1(P)} had to be bounded above by {n^{1+o(1)}}; and for decades there was significant effort expended on upper bounding {d_1(P)}, with the best known upper bound being {d_1(P) \ll n^{4/3}}, established by by Spencer, Szemerédi, and Trotter in 1984. We will note here that it seems extremely difficult to improve this upper bound. One reason for this is that if one replaces the equation (1) with the superficially similar equation

\displaystyle \mathrm{Im} w = (\mathrm{Re} w)^2 \ \ \ \ \ (2)

(i.e., replace the unit circle {(\mathrm{Re} w)^2 + (\mathrm{Im} w)^2 = 1} by a standard parabola), then the {n^{4/3}} bound is best possible, as can be seen by taking {P} to be a rectangle in the Gaussian integers of width {\asymp n^{1/3}} and height {\asymp n^{2/3}}. Hence any improvement of the {n^{4/3}} bound would have to exploit some special property of the unit circle that is not shared by the parabola.

It came as some surprise recently when a team from OpenAI resolved the question of Erdös:

Theorem 2 (OpenAI construction) There exists point sets {P} of arbitrarily large cardinality {n} such that {d_1(P) \gg n^{1 + c}} for some absolute constant {c > 0}.

The optimal value of {c} is still unknown, but the best upper and lower bounds on {c} are tracked at this page; currently we know that {0.03583\dots \leq c \leq 1/3}.

The construction in Theorem 2 is a heavily modified version of that in Theorem 1, and uses some non-trivial amount of algebraic number theory, in particular the device of Golod–Shafarevich towers of field extensions. However, it was later observed using the Mythos AI that one could get a weaker bound with less algebraic number theory, which after optimizing parameters yields the following intermediate result between Theorem 1 and Theorem 2:

Theorem 3 (Mythos construction) There exists point sets {P} of arbitrarily large cardinality {n} such that {d_1(P) \gg n^{1 + c / \log\log\log n}} for some absolute constant {c > 0}.

Furthermore, by inserting Golod–Shafarevich towers back into the Mythos construction, one can recover the full strength of Theorem 2.

These results already have a number of expositions; see for instance this article of Alon et al., or this blog post of Bloom. As an exercise for myself, I recently spent some time trying to “digest” these constructions and place them on a common footing, with an emphasis on trying to find the minimal route to either heuristically or rigorously recovering these results relying on as little algebraic number theory as possible. The post here is a writeup of this exercise. (Disclosure: AI tools were useful for providing initial summaries of these arguments, as well as on explaining various fundamentals of algebraic number theory to me.)

The first (trivial) observation is that one can use rescaling to replace the unit distance by any other fixed distance. In particular, for any positive real {m}, if we let {d_m(P)} denote the number of pairs {p,q} whose displacement {w = q-p} obeys the equation

\displaystyle w \overline{w} = m, \ \ \ \ \ (3)

then it is clear that any construction of a point set {P} with a given value of {d_m(P)} can be rescaled to another point set {m^{-1/2} \cdot P} of the same cardinality with the corresponding value of {d_1(P)}. It turns out to be convenient to work with values of {m} that are asymptotically large, for instance the product of several large primes.

All the constructions of good point sets {P} basically involve taking all the elements of a certain ring {B} of algebraic integers up to some height. In the original construction of Erdös, {B} was chosen to be the ring of Gaussian integers {{\bf Z}[i]}, but in fact any ring of integers in a non-trivial bounded degree field extension of {{\bf Q}} would suffice to recover Theorem 1 (though always with the constant {c} not exceeding {\log 2}). To go beyond this, one has to start considering number fields of unbounded degree. As it turns out, the field extensions arising from Golod–Shafarevich towers are the most efficient for this purpose, and lead to Theorem 2; but one can work with the more elementary construction of number fields generated by many square roots of medium-sized primes, and this suffices for the intermediate result in Theorem 3.

The numerology can be explained as follows. Take {B} to be a ring of integers in some number field of degree {D}, and suppose for sake of argument that {m} is the product of {t} (rational) primes {p_1,\dots,p_t}, which for simplicity we will assume to all have comparable magnitude, thus {p_i \asymp T} for some {T} and all {i=1,\dots,t}. Thus, {m} is roughly of the size of {T^t}. In practice one wants to impose some additional “splitting” conditions on these primes {p_1,\dots,p_t}, but the prime number theorem, as well as variants such as the Chebotarev density theorem, suggest that we should be able to keep {T} reasonably close to {t} in size; for instance, if we select primes greedily then we can have {T \asymp t \log t}. In particular we expect to have {\log T \asymp \log t} in practice.

By construction, {m} splits into the product of {t} rational primes. Moving up to the degree {D} extension, one can optimistically hope that {m} splits further into the product of {Dt} primes in {B}. Using conjugation symmetry, these primes might split into {Dt/2} conjugate pairs {\rho, \overline{\rho}}. By selecting one element from each pair and multiplying, this generates {2^{Dt/2}} solutions to (3) in {B}. These solutions {w} will of course have complex magnitude {m^{1/2}}; one can optimistically hope that they in fact have “height” {O(m^{1/2})} in some sense.

To take advantage of this, take {P} to be the set of points in {B} of height {O(m^{1/2})}. As {B} has rank {D}, we therefore expect the size {n = |P|} of this set to be roughly

\displaystyle n \approx (m^{1/2})^D \approx T^{D t / 2}. \ \ \ \ \ (4)

(For this heuristic discussion I will be deliberately vague about what the symbol {\approx} means.) Meanwhile, using our solutions to (3), we expect to have

\displaystyle d_m(P) \gtrapprox n 2^{Dt/2} \approx n^{1 + \frac{\log 2}{\log T}} \approx n^{1 + \frac{\log 2}{\log t}}. \ \ \ \ \ (5)

But this can be clarified by the heuristic (4). Taking logarithms, we expect to have

\displaystyle \log n \asymp D t \log T \asymp D t \log t. \ \ \ \ \ (6)

In the regime where the degree {D} of the number field is held fixed, we thus expect {t} to exhibit logarithmic type growth in {n}, and on inserting this back into (5) we (heuristically) recover Theorem 1 (with the natural constant {c = \log 2}). In fact it is not hard to turn the above heuristics into a rigorous argument, by setting {B} equal the Gaussian integers {{\bf Z}[i]} and selecting all the primes {p_1,\dots,p_t} to be {1 \hbox{ mod } 4}, so that they split completely in {{\bf Z}[i]} by the Fermat two-square theorem.

But if one can permit the degree {D} to grow in the construction, and in particular be superpolynomial in {t}, then the above heuristics suggest that we can start improving upon Theorem 1 , and even get all the way to Theorem 2 if we can make the degree go to infinity while keeping the number of primes {t} fixed.

If one naively tries this approach by forcing all the primes {p} to split completely in a very high degree number field, one runs into significant technical difficulties, not least of which is the need to obtain good error terms in the Chebotarev density theorem, which touches upon such difficult questions as the Generalized Riemann Hypothesis and the existence of Siegel zeroes. From an algebraic number theory perspective, this is related to the breakdown of unique factorization in such number fields, as measured by the class group. The size of this group is in turn controlled by the discriminant of the field, as per the fundamental theorem of Minkowski in this subject.

But one can hope that the arguments are robust enough to tolerate a little bit of breakdown in unique factorization, so long as the class group is not too large. The most natural way to do this is to use all the standard machinery of algebraic number theory, such as the unique factorization of ideals. But there turns out to be a more elementary (though largely equivalent) approach, which is to weaken the target equation (3) to a congruence equation

\displaystyle w \overline{w} = 0 \hbox{ mod } mB. \ \ \ \ \ (7)

This condition does not pin down the value of {w \overline{w}} completely, but so long as we can keep the height of {w} not too much larger than {m^{1/2}}, it does restrict {w \overline{w}} to a sufficiently small set of possible values that a simple application of the pigeonhole principle can allow one to conclude.

In order for this strategy to work well, one needs to locate high-degree number fields {B} of controlled discriminant for which it is relatively easy to at least partially split one’s rational primes {p_1,\dots,p_t} into ideals in this field {B}. It turns out that requiring the field to have a tower structure and admit complex multiplication (which basically amounts to it including {i}) is already sufficient to get a satisfactory amount of splitting. To control discriminants, the most efficient choices are the Golod–Shafarevich towers, for which the (root) discriminant stays bounded; but a more naive choice of a tower of quadratic extensions also gives reasonable control on discriminants and is sufficient to establish Theorem 3.

The three constructions thus sit on a continuum, with the key differences being the selection of the key parameters {t} (the number of primes multiplied together) and {D} (the degree). The Erdös construction keeps the degree fixed and sends the number of primes to infinity. The OpenAI construction does the opposite, keeping the set of primes fixed but sending the degree to infinity. The Mythos construction is a compromise, in which the degree and the number of primes both go to infinity in a coupled fashion. In particular, one could easily imagine an alternate timeline of events in which the Mythos construction was the first to be discovered (by either humans or AI) after the Erdos construction as a reasonably natural modification of the latter, and then subsequently refined (again either by humans or AI) to the OpenAI construction once the significance of Golod–Shafarevich towers was realized.

In this recent paper of Pohoata, the terms “horizontal amplification” and “vertical amplification” were proposed for the technique of constructing large configurations by increasing {t} and {D}, thus the Erdös construction becomes a paradigm for horizontal amplification while the OpenAI construction becomes a paradigm for vertical amplification (and the Mythos construction utilizes both types of amplification). See also this paper of Bloom-Sawin-Schildkraut-Zhelezov for another recent application of vertical amplification.

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