{"id":"f08adefa-7545-4d46-8b38-3a84cf3ccfaf","arxiv_id":"2507.14407","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"On the torus, Hausdorff dimension above a threshold 1-epsilon forces arbitrary-length polynomial progressions for distinct-degree polynomials with zero constant terms.","lead":"This paper shows that any compact subset of the circle with Hausdorff dimension sufficiently close to 1 must contain polynomial progressions x, x+P1(y), ..., x+Pk(y) for any prescribed polynomials with distinct degrees and zero constant terms, with a nonzero step y. The proof uses Sobolev smoothing inequalities and Frostman measures, and also bounds the size of the set where certain polynomial ergodic averages fail to converge pointwise.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unproved transfer of KMPW24b PET induction to the torus, plus the quoted DR24 smoothing lemma, are load-bearing; Theorem 3.1 and the degree-lowering induction collapse without them.","rationale":"The reader's weakest_assumption identifies exactly the two black-box inputs I find most load-bearing. I also checked the rest of the argument for internal gaps: the limiting construction of ν in Proposition 4.8 is only shown to have uniformly bounded total mass; uniqueness of the limit is not established, but the existence of a nontrivial weak-* subsequential limit with positive mass and support in the progression set would suffice for Theorem 1.1. Minor imprecisions (choosing s = dim_H(E) in Theorem 1.1, Frostman lemma at equality of dimension, and the U^1 notation) are harmless and easily patched. The Littlewood-Paley step in Theorem 4.1 is sound, and Proposition 4.2 follows from Theorem 1.2 by normalization. The main theorem is likely correct, but as written it depends on unverified transfers of external theorems; the verdict CONDITIONAL is appropriate and no adjustment is needed.","tokens_in":28986,"tokens_out":21335,"duration_ms":183132,"concrete_test":"Produce a fully written proof of Theorem 3.4 for T by running the PET induction of [KMPW24b, Section 6] with the torus as the ambient group and h_i ∈ R integrated against the Fejér measures ν_{[H_i]}; check that the scales H_i ≈ δ^{O(1)}N^{deg P_k} hold with constants depending only on P and that the Gowers-Cauchy-Schwarz steps do not require H_i ≤ 1. Separately, re-derive Lemma 3.9 by translating [DR24, Lemma 4] to Fourier coefficients on Z, verifying the exponent c = c(s,σ). If either proof cannot be completed with the stated hypotheses, Theorem 3.1 and the degree-lowering induction do not go through, so Theorem 1.2—and a fortiori Theorem 1.1—are unproved as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weakest point is Theorem 3.4, on which Theorem 3.1 and hence the entire Gowers-norm control rest. It is an unproved transfer of [KMPW24b, Theorem 6.10] from R to T; Remark 3.5 only says the proof is 'essentially identical' and 'can be further simplified since T is an abelian compact group.' This matters because the PET induction in [KMPW24b] is formulated for a locally compact field and uses interval scales H_i ≈ δ^{O(1)}N^{deg P_k}; on T the x-integral is compact while the h_i variables are integrated over intervals [H_i] ⊂ R, and the exact δ-dependence in these scales is needed for the passage from the box norm to the U^s norm in Lemma 3.6. If any Cauchy-Schwarz or change-of-variables step fails when H_i exceeds the period, or if the implicit constants gain an extra N^eta, the conclusion δ^{O(1)} ≲ ∥f_k∥_{U^s} of Theorem 3.1 fails. Lemma 3.9 is a second unproved input, quoted as an adaptation of [DR24, Lemma 4]; it supplies the negative-Sobolev estimate that upgrades U^{s+1} control to U^s control in Propositions 3.10 and 3.12. Both are external results, and neither is proven in the periodic setting, so the proof of Theorem 1.2 is conditional as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":29301,"tokens_out":21989,"duration_ms":234400,"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":[{"comment":"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.","section":"§3.1, Theorem 3.4 and Remark 3.5"},{"comment":"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.","section":"§3.2, Lemma 3.9"},{"comment":"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.","section":"§4.1, proof of Theorem 4.1, c ∈ (0,1) case"},{"comment":"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.","section":"§4.2, Proposition 4.8"}],"minor_comments":[{"comment":"The notation ∥f∥_{ℓ^p(T)} is used for functions on Z; it should be ∥f∥_{ℓ^p(Z)}.","section":"§2.1"},{"comment":"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.","section":"§3.1, Theorem 3.4"},{"comment":"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.","section":"§4.2, Proposition 4.8"}],"recommendation":"major_revision","confidential_remarks":"The main novelty is plausible and the paper contains a substantial amount of correct and useful technical work, but the current version is not suitable for publication without a complete proof of the torus transfer of the PET induction, a proof of Lemma 3.9, and a fix of the Frostman-convolution issue in Section 4.2. The false assertion about K_M * μ being s-Frostman is particularly serious and will require a reworking of the limiting argument in Proposition 4.8."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nNow 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.\n\nWho 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.","headline":"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.","tokens_in":29847,"tokens_out":2028,"would_cite":true,"duration_ms":444889,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B30","28A78","37A30"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["polynomial Szemerédi theorem","Hausdorff dimension","torus","Sobolev smoothing inequality","Gowers norms","Frostman measures","degree-lowering method","pointwise ergodic convergence"],"falsifier":"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.","tokens_in":28747,"feed_emoji":"📐","tokens_out":9885,"duration_ms":102978,"temperature":0.7,"pith_summary":"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.","feed_headline":"Near-full-dimensional circle sets force polynomial progressions","feed_subtitle":"Dimension alone, with no Fourier decay condition, forces all k+1 pattern points into E for some nonzero y.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the PET induction theorem that the paper transfers to the torus to get Gowers-norm control of the counting operator.","marker":"[KMPW24b]"},{"why":"Introduces the degree-lowering method that converts Gowers-norm control into Sobolev smoothing in the continuous periodic setting.","marker":"[Pel19]"},{"why":"Provides the negative Sobolev control of multiplicative derivatives that is adapted as Lemma 3.9 for the degree-lowering argument.","marker":"[DR24]"},{"why":"Sets out the Frostman-measure strategy for extracting nontrivial polynomial progressions from Sobolev estimates, which Section 4 follows.","marker":"[FGP22]"},{"why":"Establishes the precedent that dimension alone suffices for a nonlinear two-term pattern, motivating the dimension-only assumption here.","marker":"[KOS23]"},{"why":"Proves the quadratic Roth configuration under dimension alone on $\\mathbb{R}$, the closest previous result that this paper extends to longer progressions on the torus.","marker":"[Zhu24]"}],"fun_headline_variants":["High-dimension torus sets contain polynomial progressions","Nearly full dimensional sets on torus force polynomial patterns","Dimension alone on torus yields nonzero polynomial configurations","Torus sets of dimension near 1 contain polynomial progressions","High-dimensional torus sets force polynomial progressions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["High-dimension torus sets contain polynomial progressions","Nearly full dimensional sets on torus force polynomial patterns","Dimension alone on torus yields nonzero polynomial configurations","Torus sets of dimension near 1 contain polynomial progressions","High-dimensional torus sets force polynomial progressions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000738,"raw_usage":{"total_tokens":3318,"prompt_tokens":986,"completion_tokens":2332,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":602,"completion_tokens_details":{"reasoning_tokens":2256}},"tokens_in":602,"tokens_out":2332,"duration_ms":19366,"temperature":1.0,"reasoning_tokens":2256,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:58:56.481049+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}