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Polylog-dense sets of integers and primes contain almost every random linear configuration.

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Polylog-dense subsets of [N] and of the primes contain nontrivial configurations x+b₁m,…,x+bₖm for almost every coefficient vector b in wide ranges of scales.

T0 review reviewed 2026-07-31 challenge →

load-bearing objection Solid quantitative additive-combinatorics paper: polylog density forces almost-all random translation-invariant configurations in [N], and relatively dense primes in a shorter range, via a new uniform GvN + degree-lowering to U^{1+}.

arxiv 2607.28091 v1 pith:NJS4YR76 submitted 2026-07-30 math.NT math.CO

Random linear configurations in dense sets and primes

Lasse Grimmelt , Joni Ter\"av\"ainen This is my paper
classification math.NT math.CO MSC 11B3011N1337A45
keywords linear configurationsGowers normsU^{1+} normdegree loweringgeneralised von Neumannprimesdensificationdensity increment
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

The paper shows that once a subset of the first N integers is denser than a fixed power of 1/log N, it already contains nontrivial patterns of the shape x plus b_i times m for almost every choice of the coefficient vector b of size B, across a very wide range of B. The same conclusion holds for subsets of the primes that are only polylogarithmically dense relative to the primes, though the allowed range of B is shorter. The patterns are the natural translation-invariant linear configurations in two variables; arithmetic progressions are the special case b_i = i. Because the coefficients are allowed to grow, classical Gowers-norm control loses uniformity; the authors restore it by averaging over b, then lower the resulting high-degree control all the way to the U^{1+} norm, which detects long arithmetic progressions. A densification step transfers the bounded result to the primes via a truncated sieve majorant. The upshot is a quantitative existence theorem that is far stronger than what is known for any fixed large coefficient vector.

Core claim

Every subset of [N] denser than N/(log N)^{c_k} contains a nontrivial configuration x+b_1 m,...,x+b_k m with m in [N/B] for all but an O((log N)^{-c_k}) proportion of coefficient vectors b of size B, provided (log N)^{1/c_k} ≤ B ≤ N exp(-(log N)^{c'_k}). The identical statement holds for relatively polylog-dense subsets of the primes when B is at most exp((log N)^{c_k}).

What carries the argument

A quantitative generalised von Neumann theorem that controls the averaged counting operator R_H by the U^{1+} norm: after iterated Cauchy–Schwarz and concatenation produce U^k control, degree-lowering (dual-difference interchange plus major-arc analysis of phases) reduces the degree to U^{1+}, which is strong enough for a density-increment argument.

Load-bearing premise

The transfer to the primes needs a truncated sieve majorant that obeys a two-scale linear-forms condition only when the coefficient size B stays below exp of the square root of the small-prime level; larger B creates local Euler-factor obstructions the paper does not remove.

What would settle it

Exhibit a subset of [N] denser than N/(log N)^{c} that avoids the configuration x+b_i m for a positive-density set of b in ((B/2,B])^k inside the claimed range of B, or show that the truncated GPY majorant fails the linear-forms condition for some admissible system when B exceeds exp(w^{1/2}).

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Almost every translation-invariant linear pattern in two variables appears in every polylog-dense set of integers, far beyond the range known for any fixed pattern with large coefficients.
  • The same almost-everywhere statement holds inside the primes once relative density exceeds a power of 1/log N, for coefficients up to exp((log N)^c).
  • The U^{1+} inverse theorem and the averaged concatenation estimates become available as black-box tools for other random or averaged configuration problems.
  • Density-increment arguments that previously required fixed small coefficients now run uniformly for growing random coefficients.

Where Pith is reading between the lines

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

  • If the open 'almost-all forms' pseudorandomness condition suggested in the paper can be verified, the prime result would reach the same coefficient range as the integer result.
  • The same averaging-plus-degree-lowering strategy should apply to other sparse or unbounded settings once a suitable two-scale majorant is available.
  • Quantitative bounds for almost-all configurations may be convertible into effective bounds for a positive-density set of explicit coefficient vectors by a second-moment argument.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper proves that every subset A of [N] with density at least (log N)^{-c_k} contains a nontrivial configuration x+b_1 m,...,x+b_k m (m in [N/B]) for all but an O_k((log N)^{-c_k})-proportion of coefficient vectors b in ((B/2,B]\cap Z)^k, whenever (log N)^{1/c_k}\le B\le N exp(-(log N)^{c'_k}). An analogous statement holds for relatively polylog-dense subsets of the primes, but only in the shorter range B\le exp((log N)^{c_k}). The argument proceeds by a quantitative generalised von Neumann theorem giving U^k control of the averaged counting operator R_H (Prop. 3.7), degree lowering of the dual function first to U^2 and then to the U^{1+} norm (Lemmas 4.5–4.7), densification under a two-scale linear-forms condition (Thm. 6.7), verification of that condition for a truncated GPY majorant (Prop. 7.4), and a density-increment iteration (Lem. 8.4).

Significance. The results give a polylogarithmic density threshold for almost all translation-invariant linear configurations in two variables, substantially stronger than the best known bounds for fixed configurations (Kelley–Meka/Bloom–Sisask for 3-APs, Green–Tao for 4-APs, Leng–Sah–Sawhney for longer APs). The same threshold is obtained relatively in the primes, albeit in a shorter coefficient range forced by the sieve. The technical contributions—a polynomial-loss GvN uniform in the coefficient scale B, degree lowering all the way to U^{1+}, and a carefully truncated two-scale GPY majorant—are of independent interest and are tracked with explicit polynomial dependencies throughout. The limitations of the prime range are stated honestly (Rem. 7.5).

minor comments (4)
  1. [Theorems 1.1 and 1.2] The constant c_k is used both as a density exponent and (with a different value) as a range exponent; a brief remark in the statements of Theorems 1.1–1.2 that the same symbol may stand for different positive constants depending only on k would avoid any momentary confusion.
  2. [Lemma 4.7] In the proof of Lemma 4.7 the smoothing parameter A=10 is fixed without comment; a parenthetical that any A>1 works and that 10 is chosen only for convenience would make the dependence clearer.
  3. [Introduction / Remark 7.5] Remark 7.5 already notes that an “almost-all forms” linear-forms condition might enlarge the prime range of B. A one-sentence forward reference in the introduction (after the statement of Theorem 1.2) would help the reader anticipate this limitation.
  4. [Equation (1.9)] Typographical: “Parithmetic progression” in (1.9) should be “P arithmetic progression”; a few other missing spaces appear in the same display.

Circularity Check

0 steps flagged

No significant circularity: self-contained analytic derivation from standard inequalities and external cited inputs

full rationale

The paper is a pure existence/quantitative additive-combinatorics argument. Theorems 1.1–1.2 are obtained from a written chain: iterated Cauchy–Schwarz (Lemma 3.3) plus quantitative concatenation (Lemma 3.5 / Cor. 3.6) give U^k control (Prop. 3.7); degree lowering (Lemmas 4.5–4.7, following Peluse–Prendiville) reduces to U^{1+}; densification (Thm. 6.7) and a truncated GPY majorant (Prop. 7.4, extending Green–Tao) transfer the bound to the primes; density increment (Lem. 8.4) finishes. Constants c_k, C_k are existential “sufficiently small/large” parameters, not fitted to data. Self-citations ([12], [13]) supply methodological templates (densification, relative inverse theorems) whose proofs are re-derived or adapted in-line with explicit linear-forms hypotheses; they do not force the main claims by definition or uniqueness import. No prediction is statistically forced by a fit, and no equation is equivalent to its input by construction. Honest non-finding: score 0.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 2 invented entities

The paper is a pure-math existence proof. It relies on standard analytic and combinatorial inequalities, on previously established inverse and concatenation theorems, and on the Green–Tao / GPY sieve infrastructure. No numerical parameters are fitted to data; the only free choices are the usual “sufficiently small/large” absolute constants that absorb polynomial losses. No new physical or combinatorial entities are postulated beyond ordinary counting operators and the already-known U^{1+} norm.

free parameters (1)
  • c_k, c'_k, C_k (existential density and range exponents) = existential; depend only on k
    Chosen sufficiently small or large in terms of k only, to absorb polynomial losses from Cauchy–Schwarz, concatenation, and degree lowering. Not fitted to external data.
axioms (5)
  • standard math Standard properties of Gowers norms U^s, including the recursive identity and the U² inverse theorem via Fourier analysis.
    Used throughout §§3–5; classical.
  • domain assumption Quantitative concatenation lemma in the spirit of Peluse–Prendiville (Lemma 3.5).
    Proved in the paper by a Fourier-free argument, but the strategy is taken from the cited nonlinear Roth work.
  • domain assumption Green–Tao / GPY linear-forms estimates for sieve weights, extended to two scales (Lemma 7.3, Prop. 7.4).
    The majorant construction and local-factor analysis follow Green–Tao Appendix D, with extra uniformity in coefficients of size B.
  • standard math Shiu’s bound on averages of divisor powers in arithmetic progressions.
    Invoked in Lemma 7.2 to control divisor tails after truncation.
  • standard math Vinogradov’s lemma on simultaneous small fractional parts.
    Used in the major-arc analysis inside degree lowering (Lemmas 4.5, 4.7).
invented entities (2)
  • Counting operators r_H(b; f₁,…,f_k) and R_H(λ; f₁,…,f_k) independent evidence
    purpose: Package the multilinear configuration count and its average over coefficient vectors so that inverse theorems can be stated cleanly.
    Standard multilinear averages rewritten for the paper’s notation; not a new ontological object.
  • Normalised dual function D_μ and the (K,L,η) two-scale linear-forms condition independent evidence
    purpose: Make densification and majorant verification quantitative at scales (N,B,H).
    Technical packaging of existing densification and pseudorandomness ideas; the two-scale formulation is adapted to the paper but not a new physical entity.

reviewed 2026-07-31 · how reviews work

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Cite this review

Pith. "Pith review of Random linear configurations in dense sets and primes." pith.science (2026). https://pith.science/paper/NJS4YR76

@misc{pith2026260728091,
  author       = {Pith},
  title        = {Pith review of: Random linear configurations in dense sets and primes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NJS4YR76}},
  note         = {Machine review of arXiv:2607.28091}
}
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abstract

We prove that every polylogarithmically dense subset of $[N]$ contains a nontrivial configuration $x+b_1m,\ldots,x+b_km$ for almost all choices of the coefficient vector $(b_1,\ldots, b_k)$ in a wide range of scales. We prove the same statement for polylogarithmically relatively dense subsets of the primes, in a shorter range of scales. The main ingredients are a new quantitative generalised von Neumann theorem, degree lowering to the $U^{1+}$ norm, and densification arguments that transfer the result to the primes.

discussion (0)

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Reference graph

Works this paper leans on

13 extracted references · 3 linked inside Pith

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This paper was first reviewed by grok-4.5 on July 31, 2026.