REVIEW 2 major objections 3 minor 21 references
Lower bounds in the polynomial Szemer\'edi theorem
T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read This paper constructs subsets of $\{1,\dots,N\}$ of size $N^{0.7685}$ with no nontrivial triple $\{x,x+y,x+y^2\}$, surpassing the $N^{3/4}$ ceiling of the classical square-difference construction.
desk verdict A solid lower-bound paper; the flagged finite check passes and the Theorem 1.6 worry is a misreading. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is a sequence of digit sets satisfying the modular obstruction property $(R_{n+1}-R_{n+1})\cap f(R_n-R_n)\subseteq\{0\}\pmod m$. Starting from the $m$-ary expansion (base-$m$ representation) of an integer, one restricts the digit in position $i$ to $R_n$ when $k^n$ exactly divides $i$, with $R_0$ the full residue set. If a forbidden configuration existed, comparing the first differing digit would force the lowest differing position of the polynomial term to be exactly $k$ times that of the linear term, and the resulting digit congruence would contradict the obstruction property; counting how many positions fall into each $R_n$ gives the exponent $\gamma$. The multivariate version works in base $m^k$ with a single digit set $R'$ whose nonzero differences avoid $F(\mathbb Z^n)\pmod{m^k}$, exploiting homogeneity to lift a local root condition to every digit position.
What would settle it
Run the modular check for both adjacent pairs in the periodic pattern: with $R_0=\{0,\dots,64\}$, verify that no non-zero residue lies in $(R_1-R_1)\cap\{(r-r')^2\bmod 65:r,r'\in R_0\}$, and no non-zero residue lies in $(R_2-R_2)\cap\{(r-r')^2\bmod 65:r,r'\in R_1\}$. If either intersection contains a non-zero residue, the construction behind Theorem 1.1 is invalid.
Extended reading notes
Core claim
The paper establishes that modular digit constructions can be iterated: instead of a single allowed digit set whose differences avoid a polynomial value modulo $m$, one allows a sequence of digit sets $R_n\subseteq\{0,\dots,m-1\}$ indexed by the exact power of $k$ dividing the digit position, subject to $(R_{n+1}-R_{n+1})\cap f(R_n-R_n)\subseteq\{0\}\pmod m$. For the triple $\{x,x+y,x+y^2\}$, taking $k=2$, $m=65$, and the periodic digit choice $(R_0,R_1,R_2)$ displayed in the paper yields sets of size $N^{0.7685-\varepsilon}$, where $0.7685\ldots = \frac12+\frac{\log_{65}7}{3}+\frac{\log_{65}17}{6}$. The same scheme handles inhomogeneous univariate polynomials under a local-root condition and homogeneous multivariate polynomials in base $m^k$; in particular, choosing $m=p\equiv 3\pmod 4$ gives a set of size $N^{1/2}$ whose differences are never a sum of two squares.
Load-bearing premise
The headline construction for $\{x,x+y,x+y^2\}$ rests on an unshown finite verification that the two displayed sets of remainders satisfy the required modular condition modulo 65; if that check fails, the exponent $0.7685$ does not follow from the written proof.
Editorial extensions
If this is right
- For every $\varepsilon>0$, the construction in Theorem 1.1 gives explicit subsets of $[N]$ of size $N^{0.7685-\varepsilon}$ with no nontrivial triple $\{x,x+y,x+y^2\}$, so any future upper bound for this configuration must be at least this large.
- Theorem 1.6 applies to a wide class of inhomogeneous polynomials, including $f(x)=x^2+5x^3$, and yields sets of size $N^{0.8102}$ whose differences avoid $f(\mathbb Z)$.
- Theorem 1.7 supplies the first polynomial lower bounds for certain multivariate differences: size $N^{1/2}$ for avoiding sums of two squares and $N^{1/4}$ for avoiding sums of seven fourth powers.
- In the power-difference case, the two modular formulations are related: for square-free $m$, $r_k(m^k)\ge m^{k-1}r_k(m)$, with equality for primes not dividing $k$, so improvements in either the base-$m$ or base-$m^k$ search transfer to the other.
- These lower bounds are far above what greedy density arguments give, showing that local modular structure, not density alone, drives the current records for these configurations.
Reading between the lines
- The periodic digit-set framework leaves room for computational search: any sequence $R_n$ satisfying the modular obstruction gives an explicit exponent, so searching over larger square-free $m$ and longer periodic patterns could raise $0.7685$, a task the paper does not carry out.
- The $N^{1/2}$ sum-of-two-squares result sharpens the analogy with differences equal to a prime minus one: if an analogous local obstruction existed for $p-1$, the same method would suggest a polynomial lower bound, whereas the current record there is $N^{o(1)}$.
- The homogeneous multivariate theorem should extend to other forms satisfying the same local-root condition, such as $x^2+y^2+pxy$ with $p\equiv3\pmod4$, suggesting the $N^{1/2}$ phenomenon is a feature of the modular obstruction rather than of the specific quadratic form.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper obtains lower bounds for sets of integers avoiding polynomial configurations. The headline result, Theorem 1.1, constructs, for every epsilon>0, a subset A of [N] avoiding nontrivial configurations {x, x+y, x+y^2} with |A| >>_epsilon N^{0.7685-epsilon}, thereby exceeding the N^{3/4} barrier associated with Ruzsa's square-difference construction. The main technical framework, Theorems 1.5, 1.6, and 1.7, uses m-ary digit expansions and modular difference-set conditions to build such sets for non-linear progressions {x, x+y, x+y^k}, for univariate inhomogeneous polynomials, and for homogeneous multivariate polynomials. Applications include an N^{0.8102} lower bound for sets avoiding differences equal to x^2+5x^3, an N^{1/2} lower bound for sets avoiding differences equal to a sum of two squares, and an N^{1/4} bound for the sum of seven fourth powers. Section 5 gives a partial equivalence between the base-m and base-m^k modular constructions, including a Hensel-lifting result for prime moduli.
Significance. If the results stand, they break new ground in the polynomial Szemeredi theorem: Theorem 1.1 is the first construction above the N^{3/4} heuristic barrier for the non-linear Roth configuration, and Theorem 1.3 gives the first polynomial lower bounds for sets avoiding sums of two squares and related multivariate forms. The digit-set framework is clean and modular, and the paper contains several verifiable, parameter-free constructions. The proof of the univariate framework is mostly explicit, with size estimates displayed in closed form. The headline numerical exponent is not fitted to a target; it is computed from explicit finite digit sets. These strengths make the paper potentially valuable to the additive combinatorics and analytic number theory communities. However, as written, a load-bearing step in the proof of Theorem 1.6 contains a reversed inequality, and the proof of Theorem 1.1 contains a misprinted formula for the exponent; both are local and repairable.
major comments (2)
- [Section 3, proof of Theorem 1.6, Case 3] The contradiction in Case 3 is not proved as written. The chain derived in the text gives |a_d|/2 m^{jd/k} < m^Y. With X = k(Y+1)/d and X ≤ j < Y, we have jd/k ≥ Y+1, hence |a_d|/2 m^{jd/k} ≥ |a_d|/2 m^{Y+1} ≥ m^Y, since m ≥ 2 and |a_d| ≥ 1. This directly contradicts the chain. The manuscript instead asserts that Y can be chosen so that |a_d|/2 m^{Y+1} < m^Y, which is impossible for admissible parameters; the inequality should be reversed. The corrected argument preserves the stated exponent gamma, so the theorem is likely salvageable, but the proof as printed is incomplete at a load-bearing point.
- [Section 2, proof of Theorem 1.1] The displayed formula for gamma in the proof of Theorem 1.1 is incorrect. Substituting the periodic sequence (R0, R1, R2, R1, R2, ...) into (1.2) gives gamma = (k-1)/k + log_m|R1|/(k+1) + log_m|R2|/(k(k+1)). The printed expression, with log_m|R1|/k and (1+log_m|R2|)/(k(k+1)), yields a value greater than 1 for k=2 and m=65, contradicting the stated gamma = 0.7685... . Since the numerical value in Theorem 1.1 is correct, this appears to be a misprint, but it must be corrected for the proof to be verifiable.
minor comments (3)
- [Section 2, proof of Theorem 1.1] The finite verification of the sets R1 and R2 modulo 65 is only asserted with 'We may verify'. The verification is short and was independently confirmed from the listed digits, but the manuscript should include the explicit difference-set computation or provide the code, since this finite check supports the headline exponent.
- [Introduction, paragraph on Ruzsa's construction] The phrase 'surpassing the N^{3/4} limit of Ruzsa's construction' would be more precise as 'surpassing the conjectural N^{3/4} barrier for Ruzsa-type constructions', since the best proven Ruzsa-type bound cited in the paper is 0.7334... .
- [Throughout] There are several minor typographical errors, e.g. 'defintion' in the introduction and 'Szemer´edi' inconsistencies in the title page; a careful proofreading pass would improve readability.
Circularity Check
No circularity: the lower-bound constructions derive their exponents from explicit digit sets and modular checks, with no fitted prediction or load-bearing self-citation.
full rationale
The derivation chain is self-contained. Theorem 1.5 constructs A from prescribed digit sets in (2.1), proves avoidance via the modular condition (R_{n+1}-R_{n+1}) cap f(R_n-R_n) subset {0} mod m, and computes the size by an explicit count of m-ary expansions, yielding gamma from (1.2). Theorem 1.1 then chooses R0 = {0,...,64}, R1, and R2 with |R1| = 7 and |R2| = 17; substituting k = 2 and m = 65 gives gamma = 1/2 + log_65(7)/3 + log_65(17)/6, exactly the stated constant. The phrase 'We may verify' is a finite check of the explicitly listed sets, not a fitted parameter or an assumption of the conclusion. Theorem 1.6 is proved from Lemma 3.1 and the definition of R, and Theorem 1.7 from the modular root condition and homogeneity; neither imports a target exponent. There are no load-bearing self-citations, no imported uniqueness theorem, and no renaming of a known result as a new one. The skeptical observation about Case 3 of Theorem 1.6 concerns a possible inequality error independent of Y, which is a correctness issue rather than a circular reduction; it does not affect the circularity assessment of the paper's main lower-bound constructions.
Assumptions & free parameters
free parameters (3)
- m = 65 and digit sets R1, R2 (sizes 7 and 17) =
m=65; |R1|=7, |R2|=17
- Periodic digit pattern (R1, R2, R1, R2, ...) after R0 =
period 2 after initial full digit set
- m=5, R={0,2}, f=x^2+5x^3 =
m=5, R={0,2}
assumptions (3)
- ad hoc to paper The listed sets R1 and R2 satisfy the iterative difference-set conditions modulo 65 required by Theorem 1.5.
- standard math Hensel's lemma and the standard lifting of kth power residues modulo p to p^k when p does not divide k (Corollary 5.3).
- standard math Uniqueness and valuation properties of m-ary expansions for square-free m.
Cite this review
Pith. "Pith review of Lower bounds in the polynomial Szemer\'edi theorem." pith.science (2026). https://pith.science/paper/Q7CAEV5N
@misc{pith2026190806058,
author = {Pith},
title = {Pith review of: Lower bounds in the polynomial Szemer\'edi theorem},
year = {2026},
howpublished = {\url{https://pith.science/paper/Q7CAEV5N}},
note = {Machine review of arXiv:1908.06058}
}
abstract
We construct large subsets of the first $N$ positive integers which avoid certain arithmetic configurations. In particular, we construct a set of order $N^{0.7685}$ lacking the configuration $\{x,x+y,x+y^2\},$ surpassing the $N^{3/4}$ limit of Ruzsa's construction for sets lacking a square difference. We also extend Ruzsa's construction to sets lacking polynomial differences for a wide class of univariate polynomials. Finally, we turn to multivariate differences, constructing a set of order $N^{1/2}$ lacking a difference equal to a sum of two squares. This is in contrast to the analogous problem of sets lacking a difference equal to a prime minus one, where the current record is of order $N^{o(1)}.$
Reference graph
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