Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

On weighted multilinear polynomial averages in finite fields

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

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read When a weight's low-degree polynomial-phase sums decay by a power of $p$, weighted polynomial averages in finite fields converge to the unweighted average with error $O(p^{-c})$ — enough to count rational-function progressions of any…

desk verdict The weighted u^{d+1}-norm control in the main theorem is a real advance, but the rational-function application is overclaimed as written and false for constant φ; both are fixable. read the letter →

arxiv 2507.14414 v1 pith:Q3NEMSJJ submitted 2025-07-18 math.NT

Guo-Dong Hong This is my paper · ORCID
classification math.NT MSC 11B3011T23
keywords weightedmultilinearaveragespolynomialSzemereditheoreminfinitefieldsu^s-normcontrolrationalfunctionprogressionsexponentialsumsboxnormquantitativeL2convergence
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 studies averages of the form $\mathbb{E}_{y\in\mathbb{F}_p}\theta(y)\prod_{i=1}^k f_i(x+P_i(y)v_i)$ over the finite field $\mathbb{F}_p$, where the $P_i$ are linearly independent polynomials of degree at most $d$ and the $v_i$ are nonzero vectors. The main result, Theorem 1.2, states that whenever the weight $\theta$ is 1-bounded and has power-saving decay of all its polynomial-phase exponential sums of degree at most $d$, this weighted average differs from the unweighted product $\mathbb{E}_y\theta(y)\prod_{i=1}^k\mathbb{E}_{n_i}f_i(x+n_i v_i)$ by an $L^2$ error whose square is $O_{P,\theta}(p^{-c})$, uniformly over all 1-bounded functions. Because the error is a power of the field size, the convergence statement becomes quantitative counting machinery: it yields a weighted multidimensional polynomial pattern-counting identity (Theorem 1.3), and — via a classical exponential-sum bound for rational phases — an asymptotic formula for the rational-function progressions $x, x+P_1(\varphi(y))v_1,\dots,x+P_k(\varphi(y))v_k$, whose density consequence is that every subset of $\mathbb{F}_p^D$ of size at least $C p^{D-c}$ contains such a configuration for any nonzero rational $\varphi$. A sympathetic reader should care because the uniformity hypothesis is one checkable property of the weight alone, and the rational-function application reaches configurations of arbitrary length, beyond the three-term cases accessible by earlier methods.

What carries the argument

The carrying object is the $u^s$-norm, the supremum of $\bigl|\mathbb{E}_{y\in\mathbb{F}_p}\theta(y)e_p(-P(y))\bigr|$ over integer polynomials $P$ of degree at most $s-1$, together with Proposition 3.2, which asserts that for the generalized weighted operators $\mathcal{G}^\theta_{l,k}(x)=\mathbb{E}_y\theta(y)\prod_{i=1}^l f_i(x+P_i(y)v_i)\prod_{i=l+1}^k e_p(P_i(y)\xi_i)$, one has $\|\mathcal{G}^\theta_{l,k}\|_{L^2(\mathbb{F}_p^D)}^{O_P(1)}\le\|\theta\|_{u^{d+1}(\mathbb{F}_p)}+O_P(p^{-c_l})$, uniformly in the phases $\xi_i$. The proof alternates two steps: a box-norm control lemma bounds the $L^2$ norm of the operator by the box norm of an associated dual function $F^\theta_{l,k}$, and a degree-lowering induction expresses the dual function's directional frequency coefficient along $v_l$ in terms of the $(l-1)$-term operator, so the induction hypothesis applies without ever raising the degree threshold $d+1$. The base case is a direct frequency-orthogonality computation, and at every stage the unweighted multidimensional polynomial pattern-counting theorem is used as a black box. The design choice that makes arbitrary $k$ work is that the control is entirely by the $u^{d+1}$-norm of the weight, not by a uniformity norm of the functions $f_i$.

What would settle it

Take a genuinely fractional rational function, for instance $\varphi(y)=y/(y^2+1)$ over $\mathbb{F}_p$, fix a nonzero polynomial $P$ with zero constant term, and compute the maximum over $\xi\in\mathbb{F}_p$ of $p^{-1/2}\bigl|\sum_{y\in\mathbb{F}_p}e_p(\xi P(\varphi(y)))\bigr|$ for a growing sequence of primes; if this maximum tends to infinity, the uniform exponential-sum premise behind Proposition 1.4 fails and the rational-function corollary needs extra hypotheses. For the main theorem, a directly falsifying configuration would be a strongly $(d+1)$-uniform $\theta$ and 1-bounded functions for which the $L^2$ discrepancy in Theorem 1.2 does not decay as any power of $p$.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1.2: let $\theta:\mathbb{F}_p\to\mathbb{C}$ be 1-bounded and strongly $(d+1)$-uniform, meaning $\|\theta-\mathbb{E}\theta\|_{u^{d+1}(\mathbb{F}_p)}\le C p^{-c}$, where the $u^s$-norm is the supremum of $\bigl|\mathbb{E}_y\theta(y)e_p(-P(y))\bigr|$ over integer polynomials $P$ of degree at most $s-1$. Then for every 1-bounded $f_1,\dots,f_k$ and every sufficiently large prime $p$, $\mathbb{E}_x\bigl|G^\theta_k(x)-\mathbb{E}_y\theta(y)\prod_{i=1}^k\mathbb{E}_{n_i}f_i(x+n_i v_i)\bigr|^2=O_{P,\theta}(p^{-c})$, where $G^\theta_k(x)=\mathbb{E}_y\theta(y)\prod_{i=1}^k f_i(x+P_i(y)v_i)$. The same estimate yields Theorem 1.3, the weighted multidimensional pattern-counting identity, and, as a direct application of a classical exponential-sum bound for rational phases, Proposition 1.4: the rational-function configuration $x,x+P_1(\varphi(y))v_1,\dots,x+P_k(\varphi(y))v_k$ with nonzero $\varphi\in\mathbb{Q}(y)$ has the same asymptotic count as the unweighted independent-variable average, up to $O_{P,\varphi}(p^{-c})$ and with the average excluding poles. The intended reading is that the $u^{d+1}$-norm of the weight is the single obstruction to quantitative convergence: once its low-degree polynomial-phase exponential sums decay by a power of $p$, everything else in the average is determined.

Load-bearing premise

The load-bearing premise is that the classical bound on exponential sums with rational-function phases holds uniformly as the auxiliary frequencies range over all of $\mathbb{F}_p$, with a fixed power-saving error and constants independent of the frequencies, so that the $p^{-c}$ error of the main theorem survives the summation over all frequencies; the proof of Proposition 1.4 in Section 1.3 invokes this uniformity as a direct application rather than verifying it.

Editorial extensions

If this is right

  • Quantitative convergence: for any 1-bounded weight with power-saving decay in the $u^{d+1}$-norm, the weighted average $\mathbb{E}_y\theta(y)\prod_i f_i(x+P_i(y)v_i)$ agrees with $\mathbb{E}_y\theta(y)\prod_i\mathbb{E}_{n_i}f_i(x+n_i v_i)$ up to a fixed power of $p$ in $L^2$, uniformly over all 1-bounded functions.
  • Weighted counting identity: Theorem 1.3 asserts the weighted configuration count equals $(\mathbb{E}_y\theta(y))$ times the unweighted count up to $O_{P,\theta}(p^{-c})$; when the weight's average is bounded away from zero, the same density thresholds as the unweighted theorem apply.
  • Rational-function progressions of any length: Proposition 1.4 gives the asymptotic count for $x, x+P_1(\varphi(y))v_1,\dots,x+P_k(\varphi(y))v_k$ with nonzero rational $\varphi$, averaging away from poles, extending beyond the three-term cases treated earlier.
  • Density consequence: any subset of $\mathbb{F}_p^D$ with size at least $C p^{D-c}$ contains a nontrivial configuration of that rational-function form, with the constants depending only on the data.

Reading between the lines

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

  • Read as a criterion, the theorem says that exhibiting any concrete weight with provable power-saving $u^{d+1}$-decay — a purely exponential-sum calculation — immediately yields a quantitative weighted pattern-counting statement; the paper proves the implication and demonstrates it for rational-function phases, but does not attempt to verify the criterion for other arithmetic weights such as short-
  • A natural next step is to verify directly that the rational-phase exponential-sum bound holds uniformly over all auxiliary frequencies, since the proof of Proposition 1.4 (Section 1.3) needs the $O(p^{1/2})$ estimate to hold with constants independent of the frequencies; this verification would either complete the rational-function application as stated or delimit it.
  • The same two-step mechanism of box-norm control followed by degree-lowering should plausibly transfer to configurations with distinct rational functions $\varphi_i(y)$ in each slot, or to weights built from normalized multiplicative characters, whenever the corresponding uniform exponential-sum estimates hold; these extensions are not attempted in the paper.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies weighted multilinear polynomial averages over F_p, proving a quantitative L^2 convergence result (Theorem 1.2) under the assumption that the weight θ is 1-bounded and strongly (d+1)-uniform with respect to the u^s-norm introduced in Definition 1.1. The proof is an induction on the number of polynomial factors, combining a box-norm control lemma (Lemma 3.3) with a degree-lowering step that reduces the multilinear average to the u^{d+1}-norm of θ; the unweighted multidimensional polynomial Szemerédi theorem of Kuca is used as a black box. The paper further claims a rational-function application (Proposition 1.4) for progressions x, x+P_1(φ(y))v_1, ..., x+P_k(φ(y))v_k, stated as a direct consequence of Theorem 1.3 and Bombieri's exponential-sum bound, together with a density corollary (Corollary 1.5). The main technical core of Section 3 is coherent given Kuca's theorem, but the complex-valued formulation is not justified as written, and the rational-function application contains a substantial gap and a literal counterexample for constant φ.

Significance. If the identified issues are repaired, the paper would provide a clean quantitative criterion for L^2 convergence of weighted multilinear polynomial averages in finite fields, with an error term that is power-saving in p. The u^s-norm formulation, inspired by Teräväinen, is a useful simplification of earlier Gowers-norm approaches, and the induction in Section 3 is a coherent extension of Peluse--Kuca methods to weighted averages. The rational-function application, if proved, would go beyond the three-term rational-function results currently in the literature and would be a notable contribution. The reliance on Kuca's theorem as a black box is appropriate, and Lemma 2.2 is proved in the text. However, the missing conjugation in the complex-valued argument and the unproved uniformity in Proposition 1.4 are load-bearing issues that must be addressed before the claims are established.

major comments (3)
  1. [Section 3, Lemmas 2.2 and 3.3] The argument is written as if all functions are real-valued, but Theorems 1.2 and 1.3 are stated for 1-bounded complex-valued functions. The definition of the multiplicative derivative, △_h f(x) = f(x)f(x+h), the identity in Lemma 2.2, and the first displayed identity in the proof of Lemma 3.3, ||G^θ_{l,k}||^2_{L^2} = E_x F^θ_{l,k}(x) f_l(x), all need complex conjugates on alternating factors (or on θ(y') and f_i(x+P_i(y')v_i)) to be correct for complex-valued f. Likewise, the inner products ⟨f,g⟩ in the proof of Theorem 1.2 are written as bilinear forms without conjugation. No reduction to real and imaginary parts is stated. Since these identities drive the induction, the complex-valued theorem is not established by the text as written. This is repairable: one can insert the missing conjugates throughout, or prove the real-valued case and pass to complex functions by decomposing each f_i into real and imaginary parts.
  2. [Section 1.3, Proposition 1.4] The claim that Proposition 1.4 follows 'as a direct application of Theorem 1.3 and Bombieri's bound' is not justified. To apply Theorem 1.3, one must introduce the pushforward measure ν(z) = E^*_y 1_{φ(y)=z} on F_p and prove that ν is strongly (d+1)-uniform in the sense of Definition 1.1. This requires a uniform estimate |E^*_y ep(Q(φ(y)))| ≤ C p^{-c} for every polynomial Q of degree at most d, with C and c independent of Q and p. Bombieri's bound is normally stated for a fixed rational phase and has exceptional phases; the paper neither states a uniform version for the family Q∘φ nor rules out exceptional behavior after reducing the coefficients. Without such a uniform bound, the passage from Theorem 1.3 to Proposition 1.4 is not established, and consequently Corollary 1.5 is not proved either.
  3. [Section 1.3, Proposition 1.4] The hypothesis 'nonzero rational function' is insufficient as stated. If φ(y) ≡ 1, k = 1, P_1(y) = y, and v_1 = 1, then the left-hand side of Proposition 1.4 is E_x f_0(x) f_1(x+1), while the right-hand side is E_x f_0(x) E_n f_1(x+n). These are not equal for general 1-bounded functions, e.g. f_1 = 1_{\{0\}}. The proposition becomes plausible only after adding the assumption that φ is nonconstant (and after resolving the uniformity issue raised above). This should be fixed explicitly.
minor comments (4)
  1. [Section 3, proof of Theorem 1.2] The operator GEθ_k is used in equation (4) but is never defined; it appears to denote Eθ times the unweighted operator G^1_k, which is distinct from GEθ. Please define it explicitly to avoid confusion.
  2. [Section 2.2, Definition 2.2] The Gowers box norm should be defined with complex conjugates on the appropriate factors so that the expression is real and nonnegative; as written, for complex-valued f the quantity need not be a norm.
  3. [Section 3, Proposition 3.2] The exponent O_P(1) in the L^2-norm estimate is left unspecified; stating a concrete exponent (for instance, 16) would make the induction and the subsequent Cauchy-Schwarz steps easier to verify.
  4. [Section 1.2] There is a typo in the set notation '{P1, · · ·, Pk ∈ Z[y}' where a closing brace is missing; it should be '\{P_1, \dots, P_k\} \subset \mathbb{Z}[y]'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main theorem is a genuine implication from an explicit u^s-norm hypothesis, and the sole self-citation is contextual.

full rationale

No derivation step reduces to its own input. Theorem 1.2 assumes strong (d+1)-uniformity, an explicit u^s-norm decay bound, and concludes L2 convergence of the weighted average for all 1-bounded functions; the hypothesis is neither defined in terms of that average nor fitted to it. The proof factors through Proposition 3.2, proved by box-norm control and a degree-lowering induction, and invokes Kuca's unweighted multidimensional polynomial Szemeredi theorem (Theorem 3.1) as an external black box for 1-bounded functions; this is the theta=1 special case, not the weighted statement, so there is no circular dependence. The only self-citation [HL25] appears in the introduction as context ('In [HL25], the author and Lim developed a new method...') and is not used in the proofs of Theorem 1.2, Theorem 1.3, or Proposition 1.4. Proposition 1.4 is 'a direct application of Theorem 1.3 and Bombieri's bound'; this is not circular, but it omits the verification that the normalized pushforward of the uniform measure under phi is uniformly strongly (d+1)-uniform, i.e. that |E^*_y ep(Q(phi(y)))| decays uniformly over all Q of degree at most d. That is a missing uniformity check and a correctness risk, not an equivalence with the theorem's inputs. The central derivation is therefore self-contained conditional on Kuca's theorem and Bombieri-type bounds, with no fitted parameter renamed as a prediction.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters are fitted; the results are conditional on the strong uniformity hypothesis and on two theorems adopted as black boxes (Kuca's unweighted theorem and Bombieri's bound). No new entities are postulated.

assumptions (3)
  • domain assumption Kuca's unweighted quantitative multidimensional polynomial Szemeredi theorem (Theorem 3.1)
    Used as a black box to replace polynomial shifts by linear shifts in Lemma 3.3 and to handle the mean part in Theorem 1.2; its error term O(p^{-c}) is independent of the functions.
  • domain assumption Bombieri's bound for exponential sums with rational function phases ([BC17] Proposition 2.2)
    Supplies the strong (d+1)-uniformity of phases e_p(q(y)) with rational q, which is the only input making the weighted theorem applicable to rational function progressions in Proposition 1.4.
  • domain assumption Strong (d+1)-uniformity of the weight theta (Definition 1.1)
    All main theorems are conditional on this hypothesis; it is checked for the rational-phase weights via Bombieri's bound, but is not proven for general 1-bounded weights.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On weighted multilinear polynomial averages in finite fields." pith.science (2026). https://pith.science/paper/Q3NEMSJJ

@misc{pith2026250714414,
  author       = {Pith},
  title        = {Pith review of: On weighted multilinear polynomial averages in finite fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q3NEMSJJ}},
  note         = {Machine review of arXiv:2507.14414}
}
abstract

We study the weighted multilinear polynomial averages in finite fields. The essential ingredient is the $u^s$-norm control of the corresponding weighted multilinear polynomial averages in finite fields, which is motivated by Ter\"av\"ainen \cite{T24}. As an application, we prove an asymptotic formula for the number of the following multidimensional rational function progressions in the subsets of $\mathbb{F}_p^D$: \[ \textbf{x}, \textbf{x}+ P_1(\varphi(y))v_1,\cdots, \textbf{x}+ P_k(\varphi(y))v_k, \] where $\mathbb{V}=\{v_1, \cdots, v_{k} \in \mathbb{Z}^D\}$ is a collection of nonzero vectors, $\mathbb{P}= \{P_1, \cdots, P_{k}\in \mathbb{Z}[y]\}$ is a collection of linearly independent polynomials with zero constant terms, and $\varphi(y) \in \mathbb{Q}(y)$ is a nonzero rational function.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Simultaneous popular polynomial differences over finite fields

    math.NT 2026-07 unverdicted novelty 6.5 of 10

    Simultaneous popular polynomial differences hold over large F_p for linearly independent zero-constant polynomials, but fail for paired 3-APs over F_p^n as n grows.

Reference graph

Works this paper leans on

17 extracted references · 10 canonical work pages · cited by 1 Pith paper

  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

    2017

  2. [2]

    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

    2020

  3. [3]

    Multiple ergodic theorems for arithmetic sets

    Nikos Frantzikinakis and Bernard Host. Multiple ergodic theorems for arithmetic sets. Trans. Amer. Math. Soc. , 369(10):7085--7105, 2017

    work page 2017

  4. [4]

    Multiple recurrence and convergence for sequences related to the prime numbers

    Nikos Frantzikinakis, Bernard Host, and Bryna Kra. Multiple recurrence and convergence for sequences related to the prime numbers. J. Reine Angew. Math. , 611:131--144, 2007

    work page 2007

  5. [5]

    Polynomial averages converge to the product of integrals

    Nikos Frantzikinakis and Bryna Kra. Polynomial averages converge to the product of integrals. volume 148, pages 267--276. 2005. Probability in mathematics

    2005

  6. [6]

    Convergence of polynomial ergodic averages

    Bernard Host and Bryna Kra. Convergence of polynomial ergodic averages. volume 149, pages 1--19. 2005. Probability in mathematics

    work page 2005

  7. [7]

    Three T erm R ational F unction P rogressions in F inite F ields

    Guo-Dong Hong and Zi Li Lim. Three T erm R ational F unction P rogressions in F inite F ields. Int. Math. Res. Not. IMRN , (10):rnaf118, 2025

    2025

  8. [8]

    Lacey, and Fan Yang

    Rui Han, Michael T. Lacey, and Fan Yang. A polynomial R oth theorem for corners in finite fields. Mathematika , 67(4):885--896, 2021

    work page 2021

Show all 17 references
  1. [9]

    Multidimensional polynomial patterns over finite fields: bounds, counting estimates and G owers norm control

    Borys Kuca. Multidimensional polynomial patterns over finite fields: bounds, counting estimates and G owers norm control. Adv. Math. , 448:Paper No. 109700, 61, 2024

    2024

  2. [10]

    Multidimensional polynomial S zemer\'edi theorem in finite fields for polynomials of distinct degrees

    Borys Kuca. Multidimensional polynomial S zemer\'edi theorem in finite fields for polynomials of distinct degrees. Israel J. Math. , 259(2):589--620, 2024

    2024

  3. [11]

    A. Leibman. Convergence of multiple ergodic averages along polynomials of several variables. Israel J. Math. , 146:303--315, 2005

    2005

  4. [12]

    Uniform nonlinear szemer ' \ e \ di theorem for corners in finite fields

    Zi Li Lim. Uniform nonlinear szemer ' \ e \ di theorem for corners in finite fields. arXiv preprint arXiv:2501.04887 , 2025

    2025 arXiv

  5. [13]

    Three-term polynomial progressions in subsets of finite fields

    Sarah Peluse. Three-term polynomial progressions in subsets of finite fields. Israel J. Math. , 228(1):379--405, 2018

    2018

  6. [14]

    On the polynomial S zemer\'edi theorem in finite fields

    Sarah Peluse. On the polynomial S zemer\'edi theorem in finite fields. Duke Math. J. , 168(5):749--774, 2019

    2019

  7. [15]

    a v \"a inen. Pointwise convergence of ergodic averages with m

    Joni Ter \"a v \"a inen. Pointwise convergence of ergodic averages with m " obius weight. arXiv preprint arXiv:2401.03174 , 2024

    2024 arXiv

  8. [16]

    Miguel N. Walsh. Norm convergence of nilpotent ergodic averages. Ann. of Math. (2) , 175(3):1667--1688, 2012

    2012

  9. [17]

    Wooley and Tamar D

    Trevor D. Wooley and Tamar D. Ziegler. Multiple recurrence and convergence along the primes. Amer. J. Math. , 134(6):1705--1732, 2012

    2012

Pith tools

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