Polynomial Szemer\'edi for sets with large Hausdorff dimension on the Torus

Guo-Dong Hong

REVIEW 4 major objections 3 minor 30 references

Polynomial Szemer\'edi for sets with large Hausdorff dimension on the Torus

T0 review · 4 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For any finite family of polynomials with distinct degrees and no constant term, every compact subset of the circle with Hausdorff dimension above a small threshold must contain a nontrivial polynomial progression.

desk verdict A genuine new result—first arbitrary-length polynomial progression theorem for fractal subsets of the circle—but the proof as written rests on two unproved imported transfers that need to be supplied before the paper is fully verifiable. read the letter →

arxiv 2507.14407 v1 pith:KXAWQ4KP submitted 2025-07-18 math.CA

Guo-Dong Hong This is my paper · ORCID

classification math.CA MSC 11B3028A7837A30

keywords polynomialSzemeréditheoremHausdorffdimensiontorusSobolevsmoothinginequalityGowersnormsFrostmanmeasuresdegree-loweringmethodpointwiseergodicconvergence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that if a compact subset of the circle has Hausdorff dimension sufficiently close to 1, then it contains a nontrivial polynomial progression of any prescribed finite length, provided the polynomials have distinct degrees and zero constant terms. The saving $\epsilon$ depends only on the chosen family of polynomials, not on the set. The proof centers on a quantitative Sobolev smoothing inequality for the averaging operator that counts such progressions, and the same inequality yields a second result: the set of points where certain polynomial multiple ergodic averages fail to converge has Hausdorff dimension strictly less than 1. If the proof is correct, this gives the first arbitrary-length polynomial configuration theorem for fractal subsets of the circle under a dimension-only hypothesis, with no Fourier decay or pseudorandomness assumption.

What carries the argument

The engine is the Sobolev smoothing inequality (Theorem 1.2): for 1-bounded functions, the counting operator $\Lambda_{\mathbb{P};N}(f_0,\ldots,f_k)$ equals the product of the integrals plus $O_{\mathbb{P}}(N^{-C} \min_i \|f_i\|_{H^{-\sigma}(\mathbb{T})}^c)$. To prove it, the paper transfers the PET induction theorem of [KMPW24b] from the real line to the torus (Theorem 3.4), yielding control of the operator by a Gowers $U^s$-norm; then it runs the degree-lowering method introduced in [Pel19], with a lemma adapted from [DR24] (Lemma 3.9) controlling negative Sobolev norms of multiplicative derivatives by a higher Gowers norm. For the fractal applications, the paper upgrades the inequality to Frostman measures via a Littlewood-Paley decomposition: Lemma 4.4 shows the $L^\infty$ norm of each dyadic piece grows only like $2^{j(1-s+\tau)}$ for an $s$-Frostman measure, so the Frostman dimension $s>1-\epsilon$ converts into a saving that dominates the loss from removing the $L^\infty$ bound.

What would settle it

A direct check would be to run the claimed transfer of the PET induction on the torus for a small concrete family such as $\mathbb{P}=\{y, y^2, y^3\}$ and verify the asserted Gowers-box-norm bound with interval lengths $H_i \simeq \delta^{O(1)} N^{\deg(P_k)}$; an explicit failure there would invalidate Theorem 3.1. Alternatively, constructing a compact Cantor-type set of Hausdorff dimension larger than $1-\epsilon$ that avoids $\{x, x+y, x+P(y)\}$ for some polynomial $P$ would falsify Theorem 1.1 outright.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for every finite family $\mathbb{P} = \{P_1, \ldots, P_k\}$ of real polynomials with distinct degrees and zero constant terms, there is $\epsilon = \epsilon(\mathbb{P}) > 0$ such that any compact $E \subset \mathbb{T}$ with $\dim_H(E) > 1 - \epsilon$ contains points $x$ and $y \neq 0$ with $x, x+P_1(y), \ldots, x+P_k(y)$ all in $E$. The proof actually produces such a configuration with $y$ lying in $[N/10, N]$ for every sufficiently large $N$, exploiting the periodic averaging of the torus. The companion Theorem 1.3 states that for the continuous multiple ergodic averages $\mathcal{A}_{\mathbb{P};N}(f_1,\ldots,f_k)(x)$, the divergence set, where the limit fails to equal the product of the integrals, has Hausdorff dimension at most $1 - \epsilon$. Both results are obtained from the Sobolev smoothing inequality, Theorem 1.2, which gives a power saving $N^{-C}$ against the smallest negative Sobolev norm among the $k+1$ inputs.

Load-bearing premise

The proof presupposes that the polynomial-induction estimate proven for the real line in [KMPW24b] transfers to the circle with the same quantitative strength; the paper says the transfer is essentially identical but does not actually prove it, so if that transfer breaks, the Gowers-norm control and everything built on it fails.

Editorial extensions

If this is right

Every compact subset of the circle with Hausdorff dimension above the threshold contains the full progression $\{x, x+P_1(y), \ldots, x+P_k(y)\}$ with $y \neq 0$, and the common difference can be chosen in a fixed interval $[N/10,N]$ for large $N$.
The divergence set of the continuous polynomial multiple ergodic averages has Hausdorff dimension strictly below 1, a quantitative upgrade of almost-everywhere convergence for these averages.
Within the periodic setting, a pure large-dimension hypothesis replaces the Fourier decay condition that earlier fractal Roth theorems required.
The Sobolev smoothing inequality gives a power saving $N^{-C}$ in terms of the smallest negative Sobolev norm, which is the quantitative input used both for patterns and for convergence.
The theorem applies simultaneously to polynomials of distinct degrees, so it covers arbitrarily long patterns, not just three-term configurations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The proof's $\epsilon$ is not effective and depends on a PET induction complexity; a natural next step is to work out explicit threshold constants for small families like $\{y, y^2\}$ or $\{y, y^2, y^3\}$.
The distinct-degree assumption appears essential to the transferred PET induction as formulated; extending the result to linearly independent polynomials with repeated degrees would need a different intermediate induction step.
The construction in [Kel99] shows dimension one does not force three-term arithmetic progressions, so the dimension threshold in Theorem 1.1 cannot be pushed to the sharp value 1 for general families; the optimal $\epsilon$ is an open question.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

What you should know: this is a genuinely new theorem. Previous results (KOS23, Zhu24) only handled three-term polynomial configurations in R or R^2; here you get k-term progressions for arbitrary k on the torus under only a Hausdorff dimension assumption. The proof strategy is an honest adaptation of Peluse's degree-lowering method to the periodic setting, and the main induction is written clearly. If the theorem is right—and I think it is—it is a solid step forward.

What is good: the Sobolev smoothing inequality (Theorem 1.2) is stronger than the earlier KMPW24b version precisely because of the torus averaging, and the application to pointwise convergence divergence sets is a nice byproduct. The Littlewood-Paley argument in Section 4 (Lemma 4.4 and Theorem 4.1) is carefully done and fully proved. The inductions in Section 3 are well-structured; the base cases check out, and the dual difference interchange step is handled cleanly.

Now the soft spots. The paper has two load-bearing imports that are not proved. Theorem 3.4 is a torus transfer of KMPW24b Theorem 6.10, with Remark 3.5 saying the argument is "essentially identical" and "can be further simplified." That is not a proof. The PET induction in KMPW24b is formulated on a locally compact field; when moving to T, the x-integral is compact but the h_i variables still run over intervals [H_i] whose scales H_i ≈ δ^{O(1)} N^{deg P_k} are exactly what make Lemma 3.6 work. If any Cauchy-Schwarz or change-of-variables step fails when H_i exceeds the period, or the constants gain an extra N^eta, Theorem 3.1 and everything downstream collapse. Lemma 3.9, quoted as an adaptation of DR24 Lemma 4, is equally unproved; it supplies the negative-Sobolev control that upgrades U^{s+1} to U^s in the degree-lowering step. These are fixed-cost problems: they may be easily fixed, since the torus is often easier than R, but the author should state and prove the transfers (or give precise references with hypotheses verified) rather than leaving them as remarks.

Who this is for: people working on fractal configurations, polynomial Roth theorems, and quantitative ergodic averages. It deserves a serious referee—the main theorem is significant and the proof skeleton is credible. I would send it out and ask the referee to demand full details on Theorem 3.4 and Lemma 3.9.

Referee Report

4 major / 3 minor

Summary. The paper proves a polynomial Szemerédi-type theorem for compact subsets of the torus: if a collection of real polynomials has distinct degrees and zero constant terms, then any compact set E ⊂ T with Hausdorff dimension larger than 1 − ε(P) contains a nontrivial configuration {x, x + P_1(y), …, x + P_k(y)}. The main technical tool is a torus version of the Sobolev smoothing inequality for the averaging operator Λ_{P;N}, proved by adapting Peluse's degree-lowering method with inputs from Krause–Mirek–Peluse–Wright and Durcik–Roos. The paper also derives a quantitative Hausdorff-dimension bound for the divergence set of the corresponding pointwise ergodic averages.

Significance. If the main results are correct, Theorem 1.1 is a significant step: it gives an arbitrary-length polynomial progression theorem for fractal subsets of the circle under a dimension-only assumption, and Theorem 1.3 is a genuinely quantitative pointwise-convergence statement. The paper is well organized, states parameter dependencies explicitly, and contains self-contained proofs of several useful auxiliary facts, including the box-norm comparison in Lemma 3.6 and the Littlewood–Paley estimate in Lemma 4.4. The overall induction skeleton of Section 3 is coherent and the base cases are checked carefully. However, the proof as written relies on several unproved transfers and on at least one assertion that is false as stated, so the main theorem is currently conditional.

major comments (4)

[§3.1, Theorem 3.4 and Remark 3.5] Theorem 3.4 is load-bearing for the entire Gowers-norm control: it is used in the proof of Theorem 3.1 to obtain the bound δ^{O(1)} ≲ ∥f_k∥_{□^s_{[H_1],…,[H_s]}}, and Theorem 3.1 is in turn used for the third assumption of Proposition 3.12. The paper does not prove the torus transfer of [KMPW24b, Theorem 6.10]; Remark 3.5 only says the argument is 'essentially identical' and can be simplified. This is not sufficient for a journal proof, especially because the intervals [H_i] can have length larger than the period of T and the δ-dependence of the scales H_i is needed for the subsequent comparison with the U^s norm in Lemma 3.6. A complete proof or a precise statement with the exact parameter dependencies must be supplied.
[§3.2, Lemma 3.9] Lemma 3.9 is quoted as an adaptation of [DR24, Lemma 4] but is not proved. It is used twice, in the estimates (13) and (26), to pass from averages of negative Sobolev norms of multiplicative derivatives to a Gowers U^{s+1} or U^s norm. This step is essential in the degree-lowering method and in the proof of Proposition 3.10. The adaptation from R to T is not automatic: the constant c = c(s, σ) and the exponent on the Gowers norm must be verified in the periodic setting. The lemma should either be proved in the paper or a complete reference with the matching statement should be given.
[§4.1, proof of Theorem 4.1, c ∈ (0,1) case] The final displayed estimate in the c ∈ (0,1) case is not justified by Hölder's inequality. The paper claims that ∑_{j=0}^∞ 2^{-jcσ/8} ∥Π_j μ∥^c_{H^{-σ/4}} ≲ ∥μ∥^c_{H^{-σ/4}}, but in general ∥a∥_{ℓ^c} ≥ ∥a∥_{ℓ^2} for c ∈ (0,1), so the ℓ^c sum is controlled by the ℓ^2 norm in the opposite direction. Without an additional decay estimate for ∥Π_j μ∥_{L^2} coming from the Frostman condition, the displayed inequality is false. This is a load-bearing gap in the proof of Theorem 4.1, which is the key ingredient for Theorem 1.1.
[§4.2, Proposition 4.8] The proof of Proposition 4.8 asserts that for any s-Frostman measure μ, the convolution K_M * μ is also an s-Frostman measure. This is false for s < 1: for an interval of radius r < 1/M, the measure (K_M * μ)(B(x,r)) can be of size r M^{1−s}, which exceeds r^s as r → 0. Since Proposition 4.6 and Remark 4.7 are invoked for the measures K_M * μ, the uniform estimate (43) and the construction of the limiting measure ν(μ) are not established. The support assertion (45) and the conclusion of Theorem 1.1 therefore depend on a repair of this step, for example by a limiting argument that avoids claiming K_M * μ is Frostman.

minor comments (3)

[§2.1] The notation ∥f∥_{ℓ^p(T)} is used for functions on Z; it should be ∥f∥_{ℓ^p(Z)}.
[§3.1, Theorem 3.4] The statement of Theorem 3.4 introduces a parameter δ but the theorem as stated has no dependence on δ in the conclusion other than through the hypothesis |Λ_{P;N}| ≥ δ; this is an artifact of the formulation, but the statement could be made cleaner by writing δ implicit in the conclusion.
[§4.2, Proposition 4.8] The notation (K_M * μ)′ and (K_M * μ)′′ is introduced without definition; the intended meaning is the modulation of K_M * μ by an exponential factor, and this should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems follow from external PET induction and degree-lowering inputs, and the sole self-citation is not load-bearing.

full rationale

I walked the derivation chain and found no circular step. Theorem 1.1 is deduced from Theorem 4.1, which is deduced from Proposition 4.2, which is a corollary of the Sobolev smoothing inequality Theorem 1.2. The proof of Theorem 1.2 is an induction whose base case k=1 is proved directly via Parseval and van der Corput (Lemma 2.2), and whose induction step uses Proposition 3.12 and Proposition 3.10; these in turn rely on Theorem 3.1 and Lemma 3.9. Theorem 3.1 is proved from Theorem 3.4, explicitly quoted as [KMPW24b, Theorem 6.10], and Lemma 3.9 is quoted as an adaptation of [DR24, Lemma 4]. Both are external results not by the present author, and neither is derived from the paper's own conclusions. The paper's Remark 3.5 states that Theorem 3.4 follows by 'an essentially identical argument' to [KMPW24b, Theorem 6.10], with simplifications on the torus; this is an unproved transfer and a genuine correctness risk if the parameter dependencies fail in the periodic setting, but it is not circularity because the input is an external theorem, not the paper's target claim. The only self-citation, [HL25], appears in the introduction as background on finite-field rational function progressions and plays no role in the proofs of Theorems 1.1, 1.2, or 1.3. No fitted parameters are renamed as predictions, no quantity is defined in terms of the result it is used to prove, and no known empirical pattern is re-labeled as a derivation. I therefore assign score 0, with the caveat that the unproved torus transfer of the PET induction should be examined as a correctness issue rather than a circularity issue.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The proof introduces no new physical or mathematical entities. It does introduce several existential constants (epsilon, sigma, c, C, tau) whose values are chosen during the proof and are not explicitly computed. The mathematical content rests on standard harmonic analysis and on two imported theorems from the recent literature, listed as domain assumptions. No empirical data or fitting is involved.

free parameters (5)

epsilon(P) = not explicit
Dimension threshold in Theorem 1.1; chosen in the proof of Theorem 4.1 as a small constant depending on the Sobolev exponents c and sigma. No numerical value is given.
sigma(P) = not explicit
Negative Sobolev index in Theorems 1.2 and 4.1; exists via induction and depends on P.
c(P) = not explicit
Power in the Sobolev smoothing inequality; taken in (0,1] by reduction; appears in the condition defining epsilon.
C(P) = not explicit
Decay exponent in N^{-C} in Theorems 1.2, 4.1, and 5.2; depends on P.
tau = c*sigma/(8*(1-c))
Small parameter chosen in the proof of Theorem 4.1 to absorb the Littlewood-Paley growth in Lemma 4.4; ad hoc to the proof.

assumptions (6)

standard math Frostman's lemma and Frostman measure properties
Used in Section 2.4 to convert the Hausdorff dimension condition into the existence of an s-Frostman measure supported on E.
standard math Gowers uniformity norm properties (monotonicity, Gowers-Cauchy-Schwarz, U^2 Fourier formula)
Used throughout Section 3 for the degree-lowering method and PET induction.
standard math van der Corput oscillatory integral estimates
Lemma 2.2; used in Lemma 3.11 and base cases to get the N^{-C} decay.
domain assumption KMPW24b Theorem 6.10 (PET induction)
Imported as Theorem 3.4; provides the Gowers U^s box-norm control that underlies Theorem 3.1. The adaptation to the torus is asserted but not fully proved.
domain assumption DR24 Lemma 4 (Sobolev smoothing of Gowers derivatives)
Quoted as Lemma 3.9; controls the H^{-sigma} norm of multiplicative derivatives and is essential in the degree-lowering step (Proposition 3.12).
standard math Riesz representation theorem, Borel-Cantelli lemma, Littlewood-Paley decomposition
Used in Sections 4 and 5 to construct the measure nu, to handle lacunary subsequences, and to decompose Frostman measures into frequency pieces.

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Cite this review

Pith. "Pith review of Polynomial Szemer\'edi for sets with large Hausdorff dimension on the Torus." pith.science (2026). https://pith.science/paper/KXAWQ4KP

@misc{pith2026250714407,
  author       = {Pith},
  title        = {Pith review of: Polynomial Szemer\'edi for sets with large Hausdorff dimension on the Torus},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KXAWQ4KP}},
  note         = {Machine review of arXiv:2507.14407}
}

abstract

Let $\mathbb{P}= \{P_1, \cdots, P_{k}\in \mathbb{R}[y]\}$ be a collection of polynomials with distinct degrees and zero constant terms. We proved that there exists $\epsilon=\epsilon(\mathbb{P})>0$ such that, for any compact set $E \subset \mathbb{T}$ with dim(E)$>1-\epsilon$, we can find $y\neq 0$ so that $\{x,x+P_1(y), \cdots,x+P_k(y)\} \subset E$. The proof relies on a suitable version of the Sobolev smoothing inequality with ideas adapted from Peluse \cite{P19}, Durcik and Roos \cite{DR24}, and Krause, Mirek, Peluse, and Wright \cite{KMPW24}. As a byproduct of our Sobolev smoothing inequality, we demonstrated that the divergence set of the pointwise convergence problem for certain polynomial multiple ergodic averages has Hausdorff dimension strictly less than one.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Bourgain and M.-C

J. Bourgain and M.-C. Chang. Nonlinear R oth type theorems in finite fields. Israel J. Math. , 221(2):853--867, 2017

work page 2017

[2] [2]

Bergelson and A

V. Bergelson and A. Leibman. Polynomial extensions of van der W aerden's and S zemer\' e di's theorems. J. Amer. Math. Soc. , 9(3):725--753, 1996

1996

[3] [3]

Bourgain

J. Bourgain. A nonlinear version of R oth's theorem for sets of positive density in the real line. J. Analyse Math. , 50:169--181, 1988

work page 1988

[4] [4]

Two-point patterns determined by curves

Benjamin B Bruce and Malabika Pramanik. Two-point patterns determined by curves. arXiv preprint arXiv:2304.02882 , 2023

work page Pith review arXiv 2023

[5] [5]

Pointwise convergence of certain continuous-time double ergodic averages

Michael Christ, Polona Durcik, Vjekoslav Kovač, and Joris Roos. Pointwise convergence of certain continuous-time double ergodic averages. Ergodic Theory Dynam. Systems , 42(7):2270--2280, 2022

work page 2022

[6] [6]

Two bipolynomial R oth theorems in R

Xuezhi Chen, Jingwei Guo, and Xiaochun Li. Two bipolynomial R oth theorems in R . J. Funct. Anal. , 281(2):Paper No. 109024, 35, 2021

work page 2021

[7] [7]

A polynomial R oth theorem on the real line

Polona Durcik, Shaoming Guo, and Joris Roos. A polynomial R oth theorem on the real line. Trans. Amer. Math. Soc. , 371(10):6973--6993, 2019

work page 2019

[8] [8]

Improved estimates for polynomial R oth type theorems in finite fields

Dong Dong, Xiaochun Li, and Will Sawin. Improved estimates for polynomial R oth type theorems in finite fields. J. Anal. Math. , 141(2):689--705, 2020

work page 2020