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On polynomial progressions via transference

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For any fixed polynomial P with P(0)=0, sets in [N] avoiding x, x+a2P(y), ..., x+atP(y) are smaller than N divided by a triply logarithmic factor, with stronger bounds in special cases.

desk verdict Theorem 1.1 has a concrete error in Lemma 2.9's integral comparison, so the main integer result does not go through as written; the finite-field framework is still promising. read the letter →

arxiv 2506.13010 v1 pith:3UDK6FER submitted 2025-06-16 math.NT math.CO

Daniel Altman , Mehtaab Sawhney This is my paper · ORCID
classification math.NTmath.CO MSC 11B3011L07
keywords polynomialSzemeréditheoremtransferenceGowersnormsGowers-PelusenilsequencessupersaturationW-trickquantitativedensitybounds
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

The paper proves quantitative bounds for the polynomial Szemerédi theorem when the common difference is a fixed polynomial. Specifically, for any $P(y)\in\mathbb{Z}[y]$ with $P(0)=0$ and any distinct integers $a_1,\ldots,a_t$, every subset $A\subseteq[N]$ with no configuration $x, x+a_2P(y),\ldots,x+a_tP(y)$ has size at most $N\exp(-c(\log\log N)^c)$ when $t=3$ and $P'(0)\ne 0$, at most $N\exp(-c(\log\log\log N)^c)$ when $P'(0)\ne 0$, and at most $N(\log\log\log N)^{-c}$ when $P'(0)=0$, with $c=c(\deg P,t)>0$. The interest is that previous effective results covered only special patterns such as two-term polynomial differences, homogeneous perfect-power differences, or progressions whose polynomials have distinct degrees. A sympathetic reading is that the paper establishes the first reasonable quantitative version of Szemerédi's theorem with an arbitrary fixed polynomial common difference and arbitrary length.

What carries the argument

The load-bearing object is the pair of counting operators $\Lambda_W$ and $\Lambda_{\mathrm{Model}}$ together with the algebraic notion of a transferable polynomial pattern. A pattern is transferable if every polynomial relation in its kernel system is already a relation among its homogeneous linearized parts $P_i^*(y_1,\ldots,y_d)$; this is exactly the condition needed for the nonlinear and linearized orbits on a nilmanifold to equidistribute on the same subnilmanifold. The analytic engine is an iterative Cauchy–Schwarz stashing argument: starting from a large difference between $\Lambda_W$ and $\Lambda_{\mathrm{Model}}$, each dual function is shown to have large Gowers or Gowers–Peluse norm, the quasipolynomial inverse theorem converts this into a nilsequence, and the problem becomes a comparison of two polynomial orbits on a nilmanifold. That comparison is carried out by an iterative step-down reduction (Lemma 2.9) using quantitative equidistribution results and Hensel-type lemmas showing that $P_W(y)$ and $(\varepsilon zW+1)y^{d'}$ have identical distributions modulo $W^{d-d'}$.

What would settle it

Compute the two averages in Lemma 2.9 for $P(y)=y^2-y^4$, with $W$ chosen as in (2.1) for $w=(\log N)^{1/2}$, and with a degree-2 nilsequence whose horizontal and quadratic frequencies have denominators between $w$ and $W^{O(1)}$. If for any admissible residue class the difference exceeds $\exp(-c(\log\log N)^c)$ at the stated ranges of $N$ and $W$, the central transference bound is false; a cheaper diagnostic is whether the incomplete exponential-sum estimate at the end of Step 2 remains valid when the rational denominator is exactly $W^{O(1)}$ and the residual phase is $O(W/N)$.

Watch

Extended reading notes

Core claim

The central discovery is that the nonlinear pattern $x, x+a_2P(y),\ldots,x+a_tP(y)$ can be quantitatively transferred to a linear model pattern without any degree-lowering hypothesis. The paper proves that the count $\Lambda_W(f_1,\ldots,f_t)$ of the pattern with W-normalized difference $P_W(y)=W^{-d'}P(Wy)$ differs from the model count $\Lambda_{\mathrm{Model}}(f_1,\ldots,f_t)$ with difference $(\varepsilon zW+1)y^{d'}$ by at most $\exp(-c(\log\log N)^c)$ for 1-bounded functions supported on $[N]$. Once this transference is available, density bounds follow from known supersaturation statements for arithmetic progressions with shifted square differences. Over $\mathbb{Z}/N\mathbb{Z}$ with $N$ prime, the analogous transference statement holds for every transferable polynomial pattern, with error $\exp(-c(\log\log N)^c)$, and yields density bounds $N\exp(-c'(\log\log\log N)^{c'})$.

Load-bearing premise

The whole argument assumes that, after the W-trick removes small-prime biases and a weight corrects for density, the polynomial $P(y)$ and the shifted pure-power difference $(\varepsilon zW+1)y^{d'}$ behave identically for every residue class; if that comparison fails, the transference step collapses.

Editorial extensions

If this is right

  • For every fixed $P$ with $P(0)=0$, the maximum size of a set avoiding $x, x+a_2P(y),\ldots, x+a_tP(y)$ is at most $N\exp(-c(\log\log\log N)^c)$, and at most $N\exp(-c(\log\log N)^c)$ when $t=3$ and $P'(0)\ne 0$.
  • The finite-field polynomial Szemerédi theorem holds with quantitative decay $N\exp(-c(\log\log\log N)^c)$ for every transferable polynomial pattern, covering the earlier effective finite-field cases in one framework.
  • The transference error between the original polynomial pattern and its linearized model is quantitatively negligible, so future improvements in supersaturation bounds for linear patterns automatically transfer to improved integer density bounds.
  • The W-trick and archimedean weights lose only subpolynomial factors in $N$, keeping the bounds in the reasonable regime rather than the tower-type regime typical of density-increment arguments.

Reading between the lines

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

  • The author leaves implicit that the transference error is likely self-improving: if the supersaturation input in Lemma 2.11 is improved to the conjecturally optimal arithmetic-progression bounds, the corresponding case of Theorem 1.1 improves without any change in the stashing scheme.
  • A natural extension is to test whether transferability is necessary as well as sufficient; the paper's non-homogeneous example suggests that non-transferable patterns may require genuinely new ideas beyond a better equidistribution theorem.
  • A concrete numerical check of Lemma 2.9 for $P(y)=y^2-y^4$ on a step-2 nilmanifold would reveal whether the $P'(0)=0$ bound is an artifact of the proof or reflects the true difficulty of the sign conflict between archimedean and p-adic behavior.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proves new quantitative bounds for the polynomial Szemerédi theorem. Theorem 1.1 gives, for fixed P(y) in Z[y] with P(0)=0 and distinct integers a_1,...,a_t, bounds of the form N exp(-c(log log N)^c) for t=3 and P'(0) not equal to 0, N exp(-c(log log log N)^c) for all t with P'(0) not equal to 0, and N (log log log N)^{-c} for P'(0)=0, for sets A subset [N] avoiding x, x+a_2 P(y), ..., x+a_t P(y). The proof passes through a W-tricked polynomial P_W(y), compares the counting operator Lambda_W with a model operator Lambda_Model via stashing and an inverse theorem for Gowers norms, and reduces the core comparison to a nilsequence equidistribution statement (Lemma 2.9). Over Z/NZ, Theorem 1.6 proves a transference statement for 'transferable' polynomial patterns, yielding r_P(Z/NZ) bounded by N exp(-c'(log log log N)^{c'}), and the paper introduces the notion of transferable polynomial patterns (Definition 1.4).

Significance. If fully substantiated, Theorem 1.1 is the first reasonable quantitative bound for Szemerédi's theorem with an arbitrary fixed polynomial common difference and arbitrary length, and Theorem 1.6 unifies and extends several known finite-field cases, including results conjectured by Leng. The transference framework using stashing and the algebraic reduction to nilsequence comparison is a substantial structural contribution, and the notion of transferable polynomial patterns is natural and potentially reusable. The authors are explicit that several key estimates are sketched or deferred: Lemma 2.7 ends with 'we omit the details', Lemma 2.9's second case is described as 'we only sketch it', and the deduction of Theorem 1.6 from Theorem 3.1 is omitted. These are load-bearing gaps, so the significance is conditional on completing them.

major comments (4)
  1. [Lemma 2.7, proof of (2.6)] The proof of the key convolution estimate (2.6) stops at 'the desired result follows via direct integration on the sizes of level sets of |F(Theta)|; we omit the details.' This is load-bearing: the entire reduction of Lambda_Model to Gowers-Peluse norm control in Lemma 2.7 depends on this claim, and the final polynomial dependence on delta is determined by the choice of m and the level-set measure. Please supply the missing integration argument, including the precise threshold on eta implied by the condition eta^{-O(1)} <= N^{1/2}/W, and state the resulting quantitative dependence on delta.
  2. [Lemma 2.9, Step 2] The final displayed comparison for the q_2 = 1 case asserts that the two integrals are 'intentionally' identical after a change of variables, but the calculation is not shown. The change of variables must simultaneously absorb N^{1/2}, z, W, and the weight nu(y). A direct calculation with u = N^{1/2} epsilon z W y^{d'} does appear to confirm the identity for epsilon = +1, but the case b_d < 0 and the conversion from Riemann sums to integrals (with its error term) are not discussed. Since this identity is where the N^{-Omega(1)} error budget is spent, please include the full calculation and justify all error terms.
  3. [Lemma 2.9, Step 1, second case] The treatment of the model-operator average is introduced with 'we only sketch it'. This is the half of the inductive step that produces the characters eta_j and the uniform derivative bound (2.12) for the Q(z,y) average, and it is used in every application of Lemma 2.9, hence in the proof of Theorem 1.1. Please expand this sketch into a complete argument, in particular the downward induction via the leading coefficient z times the binomial coefficient (y choose k), and the treatment of the nu(y) weight in the averaging argument.
  4. [Theorem 3.4, Eq. (3.4)] The proof asserts that transferability implies V = Psi[1] x ... x Psi[k] 'after a moment's thought' and that the denominators introduced when writing (0, ..., v_{i,j,ell}, ..., 0) as a Q-linear combination of the (v_{i,j,1}, ..., v_{i,j,k}) depend only on P and k. This identification is the algebraic heart of Theorem 3.4 and, through Theorem 3.1, of Theorem 1.6. Please provide a detailed linear-algebra proof of (3.4) and of the uniform denominator bound, since the current presentation leaves the main algebraic step to the reader.
minor comments (5)
  1. [Abstract vs. Theorem 1.1] The abstract states only the N (log log log N)^{-Omega(1)} bound, while Theorem 1.1 gives the stronger N exp(-c(log log N)^c) for t=3 with P'(0) not equal to 0 and N exp(-c(log log log N)^c) for all t with P'(0) not equal to 0. Please align the abstract with the full theorem statement.
  2. [Lemma 2.2, proof] In the case (p,d!) not equal to 1, the proof says 'we may assume without loss of generality that d < k' before writing [p^k] = p^d [p^{k-d}] + [p^d]; this decomposition requires k >= d, and the case k <= d should be handled separately. Also the lemma states c_j congruent to 0 mod p^{2d} for j >= r+1, while the proof in that case only appears to use c_j congruent to 0 mod p^d; please reconcile the statement with the proof.
  3. [References [32] and [33]] The two Manners references appear to refer to the same work: the text cites [33] for 'stashing' after citing [32] for the same concept. Please consolidate the references and choose one citation for the stashing argument.
  4. [Section 3, deduction of Theorem 1.6] The derivation of Theorem 1.6 from Theorem 3.1 is explicitly omitted ('We omit the details of this deduction because they are essentially the same as in the previous section'). Since Theorem 1.6 is a headline result, please include the supersaturation deduction or state precisely which supersaturation results are being used and how they are applied.
  5. [Definition 1.4] Definition 1.4 uses kappa(P) from Definition 1.3, but Definition 1.3 is stated for polynomials in several variables while Definition 1.4 treats P = (x + P_1(y), ..., x + P_t(y)). Please make the notational correspondence explicit so that the kernel system is understood as taken over the one-variable polynomials P_i(y).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central transference claim reduces to external inverse and equidistribution theorems, and the one 'intentionally identical' integral comparison is explicitly derived from the definition of the weight ν(y).

full rationale

The paper's central theorem, Theorem 1.1, is obtained by a transference argument whose quantitative core is Lemma 2.9, comparing nilsequence averages over PW(y) and (ε zW + 1)y^{d'}. The apparent 'intentionally identical' step in Lemma 2.9 is not circular: the displayed chain of equalities changes variables and substitutes the explicit formula for ν(y), showing the two Riemann-integral limits coincide by construction of ν(y), not by assuming the conclusion. The weight ν(y) is a designed archimedean correction, not a parameter fitted to the target quantity, and the comparison itself still requires the actual work of the step-reduction argument and the equidistribution theorem of Leng (Theorem 2.8). The proof does rely at load-bearing points on results by the authors and close collaborators — [28] for the quasipolynomial inverse theorem, [27] for supersaturation, [39] for the archimedean weight device, and [1] for Lie-algebra computations — but these are invoked as independent theorems with stated assumptions, not as renamed versions of the present claim. No uniqueness theorem is imported from the authors' own work to forbid alternatives, and no fitted input is relabeled as a prediction. The sketched or omitted details (Lemma 2.7 'we omit the details'; Lemma 2.9 Step 1 'we only sketch it') are rigor/completeness concerns, not evidence that a conclusion is equivalent to its input. Hence the circularity score is 0.

Assumptions & free parameters 1 free parameters · 4 assumptions · 1 invented entities

The proof contains no fitted constants; the only hand-chosen quantity is the W-trick schedule w=(log N)^{1/2}. The listed axioms are unproved background theorems from additive combinatorics and nilsequence theory, several of which are preprints by the same authors or close collaborators. The only new formal object is the definition of transferable patterns, used as a proof device rather than a physical entity.

free parameters (1)
  • W-trick smoothing parameter w = w = (log N)^{1/2}, W a product of small-prime powers
    A proof schedule chosen so that W is subpolynomial in N and satisfies w >= exp(log(1/delta)^{O(1)}). It is hand-chosen to make the transference argument work, not fitted to any data.
assumptions (4)
  • domain assumption Quasipolynomial inverse theorem for the Gowers U^{s+1}[N] norm (Leng-Sah-Sawhney, arXiv:2402.17994, Theorem 1.2)
    Invoked as Theorem 2.13 to turn Gowers norm lower bounds into nilsequence structure. Not proved in this paper and has author overlap with the current paper.
  • domain assumption Quantitative equidistribution and factorization theorems for polynomial nilsequences (Green-Tao [14], Leng [24], Tao-Teravainen [45])
    Used in Lemma 2.9 (via Theorem 2.8) and in Section 3.1 (Theorems 3.2 and 3.3) to compare polynomial orbits on nilmanifolds.
  • domain assumption Gowers norm control and PET induction theorems for polynomial operators (Peluse [36, Theorem 6.1] and Peluse [35, Proposition 2.2])
    Used in Lemmas 2.6, 2.7, and Theorems B.3 and B.4 to convert large polynomial averages into lower bounds on Gowers or Gowers-Peluse norms.
  • domain assumption Supersaturation lower bounds for linear and homogeneous patterns (Kelley-Meka [16], Green-Tao [12], Leng-Sah-Sawhney [27], Prendiville [40])
    Used in Lemma 2.11 to convert positivity of the model operator into density bounds. These results are imported, not reproved.
invented entities (1)
  • Transferable polynomial pattern (Definition 1.4)
    purpose: Defines the class of patterns P for which Theorem 1.6 proves a quantitative finite-field transference bound.
    A formal notion introduced in this paper. It has no empirical handle outside the paper; it is a proof device that organizes the algebraic kernel conditions.

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Pith. "Pith review of On polynomial progressions via transference." pith.science (2026). https://pith.science/paper/3UDK6FER

@misc{pith2026250613010,
  author       = {Pith},
  title        = {Pith review of: On polynomial progressions via transference},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3UDK6FER}},
  note         = {Machine review of arXiv:2506.13010}
}
abstract

We prove new cases of reasonable bounds for the polynomial Szemer\'{e}di theorem both over $\mathbb{Z}/N\mathbb{Z}$ with $N$ prime and over the integers. In particular, we prove reasonable bounds for Szemer\'edi's theorem in the integers with fixed polynomial common difference. That is, we prove for any polynomial $P(y)\in \mathbb{Z}[y]$ with $P(0) = 0$, that the largest subset $A\subseteq [N]$ avoiding the pattern \[x, x+P(y),\ldots, x+ kP(y)\] has size bounded by $\ll_{P,k}N(\log\log\log N)^{-\Omega_{P,k}(1)}.$

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