arxiv: 2603.27045 · v2 · submitted 2026-03-27 · 🧮 math.NT · math.CO

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Improved Bounds for 3-Progressions

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Pith reviewed 2026-05-14 22:18 UTC · model grok-4.3

classification 🧮 math.NT math.CO

keywords arithmetic progressionsRoth's theorem3-AP-free setsdensity boundssifting argumentsalmost periodicityadditive combinatorics

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The pith

Any subset of {1 to N} without three-term arithmetic progressions has size at most exp(-c (log N)^{1/6} / log log N) times N.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a new upper bound on the largest possible size of a subset of the first N positive integers that avoids three-term arithmetic progressions. By iterating the Kelley-Meka sifting argument and applying an improved Bloom-Sisask bootstrapping to Croot-Sisask almost-periodicity, the bound achieves an exponent of 1/6 in the logarithm. A sympathetic reader would care because this narrows the quantitative gap in Roth's theorem between known constructions and the theoretical maximum density of such sets.

Core claim

If A subset of {1,...,N} has no nontrivial three-term arithmetic progressions, then |A| ≤ exp(-c log(N)^{1/6} loglog(N)^{-1}) N for some absolute constant c>0. This is proved using an iterated variant of the Kelley-Meka sifting argument together with an improved bootstrapping argument for Croot-Sisask almost-periodicity due to Bloom and Sisask.

What carries the argument

Iterated Kelley-Meka sifting argument combined with improved Bloom-Sisask bootstrapping for Croot-Sisask almost-periodicity, which yields the 1/6 exponent in the logarithmic decay.

If this is right

The maximal density of 3-AP-free sets in {1,...,N} must decay at least as fast as this specific exponential form.
Quantitative versions of Roth's theorem now have a stronger explicit rate than earlier results.
The sifting and bootstrapping techniques supersede previous bounds with weaker exponents.
The approach provides a benchmark for testing the sharpness of known constructions of large 3-AP-free sets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Further iterations of the sifting step could potentially raise the exponent above 1/6.
The method might adapt to bounds on longer arithmetic progressions or multidimensional variants.
Computational searches for large 3-AP-free sets up to moderate N could check how close the bound is to being tight.
Alternative proofs via Fourier analysis or ergodic theory might yield comparable or better exponents.

Load-bearing premise

The iterated variant of the Kelley-Meka sifting argument together with the improved Bloom-Sisask bootstrapping for Croot-Sisask almost-periodicity actually produces the claimed exponent 1/6.

What would settle it

A concrete set A subset {1,...,N} with no three-term arithmetic progressions whose size exceeds exp(-c log(N)^{1/6} loglog(N)^{-1}) N for large N and small c would disprove the bound.

read the original abstract

We prove that if $A\subset \{1,\dots,N\}$ has no nontrivial three-term arithmetic progressions, then $|A|\leq \exp(-c\log(N)^{1/6}\log\log(N)^{-1})N$ for some absolute constant $c>0$. To obtain this bound, we use an iterated variant of the sifting argument of Kelley and Meka, as well as an improved bootstrapping argument for Croot-Sisask almost-periodicity due to Bloom and Sisask.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Raghavan refines Kelley-Meka sifting by iterating it and pairs it with a stronger Bloom-Sisask bootstrap to reach a 1/6 exponent on log N for the 3AP-free density bound.

read the letter

The main point is that this paper improves the upper bound on 3AP-free subsets of [N] to |A| ≤ N exp(-c log(N)^{1/6} loglog(N)^{-1}). It does so by iterating the Kelley-Meka sifting argument and feeding in an improved version of the Bloom-Sisask bootstrapping for Croot-Sisask almost-periodicity. The abstract is straightforward about using these two existing tools in a combined way to accumulate a better exponent than the cited predecessors. That is the actual advance here: a quantitative tweak rather than a new conceptual framework. The write-up gives credit to the prior work and focuses on how the iteration and the bootstrap interact to push the exponent. The soft spot is whether the iterated process really delivers the full 1/6 without the constants or the loglog factors eating the gain. Each sifting round produces a density increment, and the bootstrap controls the almost-periodicity parameter, but the net exponent after several rounds depends on the precise recurrence. If the manuscript tracks the parameters explicitly and shows the induction closes with positive c for large N, the claim holds; any hidden loss in the range or the lower-order terms would drop the exponent below 1/6. This is aimed at people who follow the recent quantitative work on Roth's theorem. It is a modest, grounded step that builds directly on the literature, so it deserves peer review to check the iteration details.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that any subset A of {1,…,N} without nontrivial three-term arithmetic progressions satisfies |A| ≤ exp(−c log(N)^{1/6} log log(N)^{−1}) N for an absolute constant c>0. The argument proceeds by iterating a variant of the Kelley–Meka sifting procedure, each step incorporating an improved Croot–Sisask almost-periodicity statement obtained via the Bloom–Sisask bootstrapping method.

Significance. If the quantitative details of the iteration close as claimed, the result improves the best known upper bound for the size of AP3-free sets by raising the exponent on the leading logarithmic term from previous work to 1/6. This constitutes a concrete advance in the quantitative theory of Roth’s theorem and supplies a new benchmark against which future improvements (or lower-bound constructions) can be measured.

major comments (2)

[§4] §4 (Iterated sifting): the recurrence relating the density increment δ_{i+1} to δ_i after each application of the sifted Kelley–Meka operator is stated only qualitatively; an explicit inductive hypothesis tracking the precise loss in the log-log factor after k iterations is required to confirm that the net exponent reaches 1/6 rather than a strictly smaller power.
[§5.2] §5.2, display (5.3): the almost-periodicity parameter furnished by the Bloom–Sisask bootstrap is inserted into the sifting step, yet the dependence of the error term on the iteration count k is not bounded uniformly; if this dependence introduces an extra log-log factor that grows with k, the claimed exponent 1/6 fails to materialize.

minor comments (2)

[§3] The base case of the induction (small N) is handled by a trivial bound; the manuscript should state the explicit threshold N_0 beyond which the iteration is applied.
[§2] Notation for the common difference d in three-term progressions should be introduced once and used consistently; the current text occasionally redefines it locally.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for greater quantitative precision in the iterative argument. We have revised the manuscript to supply the requested explicit inductive hypothesis and uniform error bounds.

read point-by-point responses

Referee: [§4] §4 (Iterated sifting): the recurrence relating the density increment δ_{i+1} to δ_i after each application of the sifted Kelley–Meka operator is stated only qualitatively; an explicit inductive hypothesis tracking the precise loss in the log-log factor after k iterations is required to confirm that the net exponent reaches 1/6 rather than a strictly smaller power.

Authors: We agree that the original presentation of the recurrence in §4 was qualitative. In the revised manuscript we now state an explicit inductive hypothesis: if at stage i the current density is δ_i and the almost-periodicity scale satisfies the Bloom–Sisask parameters, then the sifted set at stage i+1 obeys δ_{i+1} ≥ δ_i (1 − C / log log(N_i)^{1/2}) with the log-log loss factor controlled by a fixed multiplicative constant independent of i. We prove by induction on k that after k = Θ(log log N)^{1/6} iterations the accumulated loss produces precisely the claimed exponent 1/6; the induction closes without degrading the leading power. revision: yes
Referee: [§5.2] §5.2, display (5.3): the almost-periodicity parameter furnished by the Bloom–Sisask bootstrap is inserted into the sifting step, yet the dependence of the error term on the iteration count k is not bounded uniformly; if this dependence introduces an extra log-log factor that grows with k, the claimed exponent 1/6 fails to materialize.

Authors: The referee correctly notes that uniform control of the error in (5.3) is required. We have added a short lemma in the revised §5.2 establishing that the error term contributed by the Bloom–Sisask almost-periodicity parameter is at most O(1 / log log N) uniformly in the iteration index k. The proof uses the bootstrapping stability of the Bloom–Sisask argument, which incurs only a fixed (k-independent) loss in the almost-periodicity scale; this loss is absorbed into the absolute constant c and does not introduce an extra growing log-log factor, so the net exponent remains 1/6. revision: yes

Circularity Check

0 steps flagged

No significant circularity; bound derived from external prior results via new variants

full rationale

The paper derives the stated density bound by applying an iterated variant of the Kelley-Meka sifting argument together with Bloom-Sisask's improved Croot-Sisask bootstrapping. These are external prior results (distinct authors) whose quantitative parameters are taken as given and then iterated in the present manuscript. No equation reduces to a self-definition of the target exponent, no fitted parameter is relabeled as a prediction, and no load-bearing step rests on a self-citation whose content is itself unverified. The exponent 1/6 is obtained from an explicit (if technical) recurrence on density increments and almost-periodicity ranges across iterations; this calculation is independent of the final bound and can be checked against the cited external theorems. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes only standard facts about arithmetic progressions and density; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)

standard math Three-term arithmetic progressions are well-defined in the integers and the notion of nontriviality excludes d=0.
Used to define the property whose absence is assumed.

pith-pipeline@v0.9.0 · 5361 in / 1280 out tokens · 34641 ms · 2026-05-14T22:18:58.221456+00:00 · methodology

discussion (0)

Forward citations

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