An AI's attack on progression-free sets: a ledger of what stalled

The most useful thing an unfinished proof can tell you is exactly where it breaks, and this one breaks in about a dozen specific places.

Executive Summary

• The work has two tasks. The first develops an argument for the reciprocal-sum problem. The second seeks a quasipolynomial bound for every fixed progression length, and that goal is not achieved.
• The baseline is Bloom and Sisask's three-term reciprocal-sum result. Qualitative density decay alone does not supply the summability the argument needs.
• Many approaches were tried: structured ambient sets, simultaneous frequency alignment, sparse counting lemmas, Croot-Sisask smoothing, lifting and iteration, corners, and entropy at scale p = log M. Each met a stated obstruction.
• The final calculation was left unfinished, without a restriction lemma or a sparse counting principle.
• For founders and technical leads, the lesson is about method: a ledger of named failures is a more reliable deliverable than a confident sketch.

Two tasks, and only one of them was ever claimed #

The material divides into two tasks, and it's worth keeping their status apart. The first develops an argument for the reciprocal-sum problem. The second seeks a quasipolynomial bound for every fixed progression length. The verbs matter. The first task is described as developing an argument. The second only seeks a bound. Nothing in the record says the second goal was reached, and I'll treat it as a target throughout.

The assistant doesn't start from nothing. It uses Bloom and Sisask's three-term reciprocal-sum result as a baseline. Everything after that is an attempt to push past the baseline, either toward longer progressions or toward a stronger quantitative statement. The notation is fixed early. For each k, r_k(N) is the largest size of a subset of {1, ..., N} with no nontrivial k-term progression. A set A of density α = |A|/N is the object under study, and the normalised function is f = 1_A/α.

That normalisation drives most of what follows. Define μ = 1_A/α, so that E[μ] = 1, and set M = α^-1. The logarithmic sparsity is then p = log(1/α), which is the same as log M. The sparser the set, the larger M and p become. Every obstruction below is, in one form or another, a statement about what happens as that sparsity grows.

Density decay isn't the same as summability #

The first obstacle appears at the level of the argument's architecture, before any Fourier analysis. The record says plainly that qualitative density decay alone does not supply this summability. Knowing that densities shrink as you move through scales is not enough. The reciprocal-sum problem needs the shrinkage to be quantified well enough to sum.

A tempting escape is a recurrence. The notes say that a sufficiently strong recurrence for extremal progression-free densities might yield summability across exponential scales. That sentence is hedged, and it should stay hedged. It describes what a hypothetical strong recurrence could do. It doesn't report one.

A second cost is separate from the first. The assistant identifies the accumulation of many weak correlations as a further cost. Each individual correlation can be harmless while their sum is not. This theme recurs: a step that is cheap once becomes expensive when it has to be repeated.

Retaining structure, and why three terms don't generalise #

One proposed remedy is to keep the structure around. The assistant proposes retaining structured ambient sets through successive increments. The idea, as hedged in the source, is that you shouldn't throw away the structured set you found at one stage when moving to the next.

The three-term case has a feature that longer progressions lack. The positivity mechanism useful for three-term progressions does not directly extend to longer rooted counts. This is the first of several places where the obstruction is specific to k greater than three. A later note makes the same point in general terms: longer progressions introduce further higher-order obstructions.

The record includes a tensor-product sketch on whether fixed uniformity norms can control sparse normalised functions. The assistant does not complete the sketch. I'd rather report that than imply the question is settled either way.

Aligning frequencies once rather than many times #

Sequential alignment of frequency functions is where the accumulation problem bites hardest. If each alignment costs a little, a long chain of them costs a lot. The assistant proposes aligning several frequency functions simultaneously, instead of accumulating losses through sequential alignment. The marginal note puts it bluntly: "No iterative losses if align all frequencies at once." Like the proposal it summarises, that sentence is hedged, because it describes what simultaneous alignment would buy if it worked.

The note then states what is still missing. S_h would need to be recoverable from frequencies with large correlations, at least exp(-poly). That remains a requirement, not a result. The correlations being organised are those of the pair products f(x)f(x + h), with f = 1_A/α and α = |A|/N.

Even granting that, the assistant still needs to preserve relative correlation and complete the counting argument. So the simultaneous-alignment idea has an attractive property, no iterative losses, and a list of things nobody has verified. A related approach has a related limitation. Truncating rare coherent peaks works for independent frequency clusters, but extending that argument to dependent clusters remains unresolved.

Rare empty pieces and the cost of adaptive filling #

When the argument moves from Fourier-side bookkeeping to actually counting progressions, a structural problem appears. The assistant identifies an obstruction: upper density bounds on structured pieces permit rare empty pieces. An upper bound on how much of A lies in a piece says nothing about whether any of A lies there. A piece that's rare and empty can sit inside a family of pieces that all satisfy the bound.

One response is adaptive filling, and it comes with its own bill. Adaptive filling introduces further feasibility conditions that the assistant has not proved. A majorant is a natural way to rule out the rare bad pieces. The assistant's quoted text says a majorant on all rare atoms is likely unrealistic. The remark is hedged, but it points away from the most direct fix.

This is the pattern that explains the rest of the record. A cheap argument needs a lower bound somewhere, the available tools give upper bounds, and the gap between them is occupied by rare, sparse, empty things.

Borrowing sparse counting machinery, and finding the borrowed part wasn't enough #

The assistant looked to existing sparse-counting literature for a way across. Two sources shaped the attempt.

On Conlon-Fox-Zhao densification, the assistant found that it relied on two-sided linear-forms estimates. That matters because the setting here has an upper-bound structure and not a two-sided one. A method built on two-sided control isn't available when only one side is known.

The sparse graph counting lemma of Filmus, Hatami, Hosseini, and Kelman supplied the model for asymmetric grid moments and positive-semidefinite unbalancing. So it served as a template, not as a drop-in theorem. Even so, the same difficulty returned. Upper bounds alone repeatedly encountered sparse holes. A further wrinkle makes the grid moments hard to read: large centred grid moments could reflect missing mass rather than upward concentration. A large moment looks like evidence of concentration, but it can equally be produced by mass that isn't there. That ambiguity is the sparse-hole problem seen from the moment side.

Next the assistant considered small power losses in M. It obtained no suitable counting theorem. That's a clean negative result, and it narrows the field: even allowing the argument to lose a small power of M, no suitable counting theorem was found.

Smoothing was too expensive to apply #

Croot-Sisask smoothing is a standard route to regularising a set before counting. The assistant considered it and found the required accuracy too costly in the proposed application. The method wasn't wrong. Its price exceeded what the argument could pay.

The same economics govern the other routes. For fibre arguments, fibre bounds did not multiply automatically, and constant-norm spheres obstructed the hoped-for strong decay over fixed digit alphabets. A product structure suggests that bounds on pieces should combine into a bound on the whole, and here they didn't. Attempted lifting and iteration still lost too much quantitatively.

The Ben-Sasson-Ron-Zewi approximate-duality approach was considered next. The assistant found that the contemplated iteration loses too much quantitatively. This is the same verdict as the lifting attempt, reached from a different direction, which suggests the trouble lies in iteration itself and not in any one construction. I'll hold that as an observation about the pattern, not a proven diagnosis.

One line of optimism, and the arithmetic that undercut it #

The record contains a moment of apparent progress. The assistant's quoted text asserts that products degree fluctuations are handled with polynomial precision, exclamation mark included. It's a hedged assertion, quoted as the assistant wrote it.

The assistant then noticed the problem with it. Signed expansions could have absolute mass of order 2^{p, which overwhelms polynomial accuracy. Recall that p = log M. Polynomial precision in M is a modest target, while a signed expansion with mass of order 2}p is exponentially larger. The precision is real, but the cancellation it relies on can't absorb that much mass.

I find this the most instructive entry in the ledger, because it shows the self-check working. A claim made with an exclamation mark was reopened, and the arithmetic it depended on was found wanting. That is what an honest working record looks like. The enthusiasm and the correction both sit in the same document.

Corners and boxes don't assemble into longer progressions #

A different strategy was to use a stronger known result as a stepping stone. The quasipolynomial corners bound is the relevant tool, but the resulting corners did not automatically assemble longer progressions. A corner is a two-dimensional pattern and a long progression is a one-dimensional one, and the passage between them isn't automatic.

The four-term box count presents a related difficulty. Positivity of the four-term box count did not control its exceptional diagonal. A positive count can still be dominated by degenerate configurations, and the diagonal is where those live.

The notes do offer a possible way forward, carefully hedged. Complete box norms offer a possible mechanism for whether upper counts alone can force lower progression counts. That question, whether one-sided information can force a lower bound, is the one the whole project keeps circling. Yet there's a limit on how far this extends. Negative cubic contributions obstruct a straightforward extension to progression systems.

Moving to entropy at scale p = log M #

With the direct counting routes blocked, the investigation turned toward entropy at scale p = log M, drawing on the Kelley-Meka use of high moments. The choice of scale is not arbitrary: p = log M is exactly the logarithmic sparsity defined at the outset.

The assistant's quoted text proposes using entropy instead of L2 by nonnegativity and frequency dissociativity (Chang). That's a proposal, not a result. The reasoning is that μ = 1_A/α is nonnegative, and Chang's dissociativity gives structure on the frequency side, so entropy might do the work that an L2 quantity can't.

For this to work the assistant needed a specific analytic input. The assistant sought a Fourier restriction estimate ||g_hat||_q = O(1) for bounded g and fixed q > 2, under suitable local upper bounds. This was a target. It wasn't established.

The obstacle appeared at the next step. Extracting structure from high moments lost too much mass. High moments are where Kelley-Meka-style arguments gain their strength, and here they were also where the argument shed what it needed to keep.

The calculation that came closest #

The final calculation deserves its own look, because it has both a success and a failure. It reaches a full endpoint bound under its convolution hypotheses. That's a real result, stated as such, and it carries the qualifier "under its convolution hypotheses". The bound holds when those hypotheses do.

But it fails to cross the exponent-three threshold needed by the proposed counting argument. The assistant's excerpt notes a weak l3 bound from convolution and cap, and that a strictly better exponent (below 3) is needed for interpolation. In plain terms, the calculation lands exactly on the wrong side of a line. A bound at exponent three is not enough, and interpolation needs something strictly better.

It's tempting to read a near-miss like this as "almost there". I'd resist that. The record doesn't say how far the calculation is from the threshold, and a gap that is a single exponent can still be the entire problem. What the record does say is that the assistant left the calculation unfinished, without deriving either the restriction lemma or the sparse counting principle. Those two missing pieces are the project's open debt.

What the ledger is worth #

It would be easy to dismiss this as a failed attempt. I think that reads the evidence wrongly, for a reason that applies well beyond number theory.

The document is specific. It names the obstruction at each junction: rare empty pieces, missing mass in grid moments, two-sided estimates in Conlon-Fox-Zhao, exponential-in-p signed mass, the exceptional diagonal of the four-term box count, negative cubic contributions. It distinguishes a demonstrated limitation from a hoped-for mechanism. It marks the recurrence for summability as something that might work, and the second task as something sought. It says what the assistant did not finish.

For a founder or a technical lead evaluating AI-assisted research, that is the thing to look for. A system that returns a confident, complete-looking argument is hard to audit. A system that returns a list of named failures, with the one near-success clearly marked as falling short of a threshold, can be audited line by line. Someone can pick up the exponent-three problem, or the dependent-cluster problem, and know what the starting position is.

There is also a pattern worth stating directly. Many of the failures are failures of accumulation: weak correlations that add up, sequential alignments that compound, iteration that loses too much, fibre bounds that fail to multiply. One proposed remedy targets this directly, aligning frequencies simultaneously so that no iterative loss is incurred. It's hedged, and its prerequisite, recovering S_h from frequencies with large correlations, is unproved. Still, it's the proposal in the record that responds most directly to the most frequent complaint.

The open question the record raises is therefore narrow and checkable. Can a bound strictly better than exponent three be obtained under the convolution hypotheses that already give the endpoint bound? If so, interpolation has something to work with. If not, the Fourier restriction route and the sparse counting principle, neither of which has been derived, are the places to look next.

Questions? Answers.

References #

• Thomas F. Bloom and Olof Sisask, (2020). Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions ⧉. arXiv.
• Zander Kelley and Raghu Meka, (2023). Strong Bounds for 3-Progressions ⧉. arXiv.
• David Conlon, Jacob Fox, and Yufei Zhao, (2014). A relative Szemerédi theorem ⧉. Geometric and Functional Analysis.
• Ernie Croot and Olof Sisask, (2010). A probabilistic technique for finding almost-periods of convolutions ⧉. Geometric and Functional Analysis.

Last reviewed 2026-11-01.