Polynomial configurations and pointwise averages along Piatetski--Shapiro primes

Hamed Mousavi

REVIEW

Polynomial configurations and pointwise averages along Piatetski--Shapiro primes

Hamed Mousavi

Combinatorics · Submitted 3 Nov 2026 · 25 references

Pith's verdict: Solid extension of the density-increment machinery to Piatetski–Shapiro primes, matching the (log N)^{-κ} threshold. The main technical work—the comparison estimate and its integration—looks correct. Real risk: the entire argument depends on two unpublished preprints that supply the power-saving estimates. If those hold, this is a substantial contribution.

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arXiv ID2611.04575 (version 1)

Pith ID5PITWYFE, Pith's permanent record of this paper and its review

CitationHamed Mousavi. "Polynomial configurations and pointwise averages along Piatetski--Shapiro primes." arXiv:2611.04575 (2026). https://pith.science/paper/5PITWYFE

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This paper does what it says: it takes the Krause–Mousavi–Tao–Teräväinen density-increment architecture for shifted primes and adapts it to the much sparser Piatetski–Shapiro primes, obtaining the same (log N)^{-κ} threshold. That is genuinely new, and the adaptation is not trivial—sparsity forces a nontrivial comparison estimate (Proposition 2.3) and a careful rescaling trick to keep the coefficient growth under control. The pointwise ergodic theorem for the corresponding averages is also new and follows cleanly from the same comparison. The writing is clear; the technical steps are laid out in enough detail that a specialist can follow them.

The elephant in the room is the paper's load-bearing dependence on two preprints: Shao–Sun [21] (Theorem 2.1) for polynomial-phase estimates of D_γ and Teräväinen [23] (Theorem 1.4) for the generalized von Neumann bound. Neither has completed peer review. The paper does not prove these results; it interfaces with them. The author acknowledges this honestly, and the Lean formalization [17] explicitly marks these interfaces as assumptions. If either estimate fails—or if the constants σ_d and K turn out to force an empty range for c—the arguments in Sections 2–4 collapse. This is a real risk, not a manufactured one.

A secondary soft spot: the closeness condition c < c* is stated to exist but not quantified in terms of the polynomial family. The proof shows that if 1−γ is small enough relative to the implicit constants from [21] and [23], then things work. For a reader who wants to know whether c* = 1.001 or c* = 1.1, the paper gives no explicit bound. This is not a flaw in the logic but a limitation in applicability.

Who should read this? Additive combinatorialists and analytic number theorists working on quantitative Szemerédi theorems or thin prime sets. The paper deserves a serious referee. The referee should verify that the interface to [21] and [23] is correctly stated and that no hidden dependence on the internal constants would make the required range for c empty. The author could partially address this by adding a remark that clarifies the dependence of c* on the degree d and the external constants.

My recommendation: send to review. The core adaptation is clean; the risk is not in the paper's own reasoning but in its reliance on yet-unverified external results. That is standard for work at the frontier—the field will settle whether those preprints are correct.

Referee report

Significance

If the main theorems hold (the central claims are logically sound given the external inputs), the paper makes a genuine advance. It extends polynomial Szemerédi‑type results—and the corresponding pointwise ergodic theorems—from the full set of primes or from sets of positive density to the very sparse Piatetski–Shapiro primes, which have power‑saving zero relative density among the primes. The density threshold (log N)^{-κ} matches the best known bound for shifted‑prime polynomial configurations (Krause–Mousavi–Tao–Teräväinen). The paper is clearly written and carefully orchestrates heavy external machinery. The availability of a Lean formalisation (even if not yet independently reviewed) is a noteworthy transparency measure. The results are likely to stimulate further work on recurrence along thin subsets of primes.

Major objections (4)

[§1 (Theorem 2.1, Theorem 1.4)] The entire argument depends on two unpublished preprints: Shao–Sun [21] (Theorem 2.1, the polynomial‑phase estimate for the Piatetski–Shapiro error weight) and Teräväinen [23] (Theorem 1.4, the generalized von Neumann bound). The paper does not prove these results; it merely cites them and sketches the derivation of Theorem 2.1 from [21]. While it is acceptable to rely on preprints, the central claim is conditional on the correctness of both external works. The author should either confirm that these preprints have been accepted for publication or, at minimum, provide a detailed summary of the specific constants (σ_d, K) and verify that the required inequality 1−γ < min(1,σ_d)/(8K) can be satisfied for a non‑empty interval of c. The Lean formalisation ([17]) explicitly lists these as unverified interfaces, which acknowledges the risk but does not resolve it. The referee recommends that the paper include a statement about the current status of [21] and [23] and, if possible, a self‑contained sketch of the relevant estimates that does not rely on other unpublished claims. The authors acknowledge this: “The external results and the remaining source-to-interface translations are not themselves proved in Lean.”
[§2, after (16)] The proof of Proposition 2.3 uses Lemma 2.2 to bound the Gowers norm ∥Θ∥_{u^{d+1}[L]} by O(L^{-σ_d}). Lemma 2.2 averages over the Q residue classes via orthogonality and applies Theorem 2.1. The estimate is claimed to be uniform in the polynomial phase ψ, but the averaging introduces a term a(n−1)/Q that shifts the phase by a linear term. Since linear terms are already covered by Theorem 2.1 (degree at most d), this is fine. However, the constant in the bound from Theorem 2.1 may depend on r (here r = d) but not on Q. The proof should explicitly note that the implied constant in Lemma 2.2 is allowed to depend on r and γ but is uniform in Q and M, which it does. This is acceptable, but the argument would benefit from a brief remark about the uniformity in Q.
[§3, Lemma 3.2] Lemma 3.2 assumes that A contains no configuration of the form (25) and then deduces the discrepancy hypothesis (18) for the ordinary prime weight. The proof uses Proposition 2.3, which requires Q ≤ exp((log N)^a) for some a < 1. In the iteration of Theorem 1.1 the authors later ensure that each Q_j satisfies Q_j ≤ exp((log N_j)^{2c_0}) and claim that this fits Proposition 2.3 by choosing a with 2c_0 < a < 1. Since N_j is smaller than the original N, the bound on Q_j in terms of N_j is weaker, but the argument that log N_j is still at least (log N)^{η_0} (line [b157]) ensures that the condition holds. This step is logically sound but would be clearer if the authors explicitly stated that the exponent a in Proposition 2.3 can be any fixed number in (0,1), and that the iteration guarantees Q_j ≤ exp((log N_j)^a) for a suitable a given the earlier parameter choices.
[§4, Proof of Theorem 1.2] The proof of the pointwise comparison (Proposition 4.1) uses the L^1 bound (38) and a lacunary subsequence argument. The bound (38) relies on the same closeness condition 1−γ < min(1,σ_d)/(2K). The constant η is then taken as min(1,σ_d)/(4K). The argument that the full sequence converges because the lacunary differences are o(1) uniformly in x is standard and correctly justified. No additional issues arise here beyond the dependence on the external constants.

Minor objections (7)

[Abstract] The abstract writes 'p=⌊n^c⌋∈P'; the blackboard bold 'P' is missing a backslash before 'mathbb{P}' in the source, but the meaning is clear; this should be corrected in the final version.
[§1, page 1] The phrase 'The polynomial Szemer´ edi theorem' uses an acute accent; the standard notation is 'Szemerédi' (with double acute). Consider using the proper Unicode or LaTeX code '\H{o}' to avoid confusion.
[§2, just before (6)] Write 'e^{2π i t}' instead of 'e 2πit' for clarity.
[§2, proof of Lemma 2.2] The phrase 'the phase … is a real polynomial in n of degree at most r' is correct. However, the sum over n from 2 to L+1 includes n=1 with D_γ(1)=0; the remark that we may include n=1 is helpful but could be placed in parentheses.
[§3, paragraph after (31)] The sentence 'The forbidden-configuration (2) hypothesis is preserved, since Q_j^{b_j} P_{i,Q_{j+1}}(y) = P_{i,Q_j}(Q_j^{b_j} y)' should note that the equality holds after an affine change of variables; the argument is clear but the equation could be typeset more carefully to avoid potential confusion with the indexing.
[§4, after (39)] The estimate for higher prime powers uses N^{3/2−γ} (log 2N)^2; it is claimed to be o(N) because γ>1/2 after decreasing c* if necessary. Since c* is chosen so that 1−γ is very small, γ is close to 1, so the estimate holds. This is fine, but the phrase 'after decreasing c* if necessary' appears twice; it may be clearer to state once at the beginning of the proof that c* has already been chosen small enough so that γ>1/2.
[References] Reference [9] is listed as 'arXiv:2608.19525v1 (2026)'; reference [21] as 'arXiv:2605.18676v1 (2026)'; reference [23] as 'arXiv:2401.03174v2 (2024)'. All are preprints. The author should update these to published versions if they become available before the final submission.

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The paper

The paper claims

稀疏素数参数下多项式配置仍有对数幂次密度门槛

The reading

这篇论文研究的是当配置中的差值取自极其稀疏的Piatetski–Shapiro素数减一时，整数集合中多项式配置的存在性与多重遍历平均的收敛性。Piatetski–Shapiro素数集的密度仅为普通素数的幂次稀疏（约$N^{1/c}/\log N$，$c>1$），但论文证明，只要$c$足够接近$1$，任何密度不低于$(\log N)^{-\kappa}$的整数子集必然包含形如$x, x+(p-1), x+(p-1)^2,\dots,x+(p-1)^k$的配置，其中$p$是Piatetski–Shapiro素数。这一密度门槛与普通素数参数下的最优结果相当。论文还证明，沿这些稀疏素数的多重多项式遍历平均几乎处处收敛，且极限与沿全体素数的平均相同。这意味着稀疏化并不改变极限行为，为在极低密度素数子集上的多项式回复理论提供了定量支撑。

What carries the argument

论证的核心机制是通过误差权重$D_\gamma = \Lambda_\gamma - \Lambda$将Piatetski–Shapiro素数权重转化为普通素数权重，其中$\Lambda_\gamma$是经过归一化的稀疏素数von Mangoldt函数。Shao–Sun的多项式相位估计（定理2.1）给出$D_\gamma$与任意多项式相位关联的幂次节省上界，Teräväinen定理（定理1.4）则将该上界转化为对更一般的多项式平均的控制，从而避免了显式的Gowers范数估计。在组合部分，这一比较被插入Krause–Mousavi–Tao–Teräväinen的密度增量论证中：通过一个巧妙的变量代换（$n=Qm,\,z=Qx$）处理系数增长问题，使得若集合缺失配置则导致密度在某个算术级数上提升，反复迭代直至密度超过$1$得出矛盾。在遍历部分，通过$L^1$比较和求和分部得到逐点收敛。

Extended reading notes

Core claim

论文的核心发现包含两个定理。定理1.1：对于任意一族满足常数项为零且次数各不相同的整数多项式$P_1,\dots,P_\ell$，存在$c^*>1$和$\kappa>0$，使得对所有$1<c<c^*$，任意充分大的$N$和任意满足$|A|\geq N/(\log N)^\kappa$的子集$A\subset[N]$，都存在一个Piatetski–Shapiro素数$p=\lfloor n^c\rfloor$，使得配置$x, x+P_1(p-1),\dots,x+P_\ell(p-1)$全部落在$A$中且两两不同。定理1.2：在同样的参数范围内，沿Piatetski–Shapiro素数的多项式多重遍历平均几乎处处收敛，且极限与沿全体素数的平均一致。

Load-bearing premise

c必须足够接近1，使得1-γ小于一个依赖于多项式最高次数d和外部估计中常数（σ_d来自Shao–Sun，K来自Teräväinen）的显式上界，但这个上界在论文中未完全量化，仅要求1-γ < min(1,σ_d)/(8K)。

Editorial extensions

If this is right

对于任意固定次数严格递增的多项式族，存在一个确定的$c^*$（仅依赖于最高次数），使得所有$1<c<c^*$对应的Piatetski–Shapiro素数参数都有相同的对数幂次密度门槛。
沿Piatetski–Shapiro素数的多项式遍历平均的极限与沿全体素数的平均相同，从而将已有关于普通素数的点态收敛结果（Mirek–Wan–Wright）扩充到极稀疏素数子集。
推论4.2给出相应的滞后$r$-变差估计（$r>2$），这为进一步的几乎处处收敛和最大函数有界性提供了定量工具。
定理1.1的证明本质上与具体的多项式族无关，只依赖于次数和系数有界，因此适用于所有满足常数项为零的有限个不同次数的整系数多项式。

Reading between the lines

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

定理1.2目前要求$f_i\in L^\infty$，但论文Remark 1.3指出可能通过更精细的估计推广到$f_i\in L^{p_i}$（$p_i>1$且$\sum p_i^{-1}\leq 1$）。若实现这一推广，将覆盖更多自然的时间演化系统。
论证中的密度增量仅依赖于稀疏素数集的幂次节省相位估计，这暗示对更一般的零密度稀疏素数集（如素数的小区间子集或Maynard的有限数字素数）也可能获得类似结论，只要存在相应的Shao–Sun型估计。
Lean形式化（[17]）仅在有限个外部输入条件下验证了推理链，但外部结果（如Shao–Sun、Teräväinen）自身尚未形式化，因此完整的计算机验证仍依赖于这些算术输入的可靠翻译。

Original abstract

For every $c>1$ sufficiently close to $1$, we prove that there is $\kappa>0$ such that every sufficiently large $A\subset[N]$ of density at least $(\log N)^{-\kappa}$ contains \[ x,\ x+(p-1),\ x+(p-1)^2,\ldots,\ x+(p-1)^k, \qquad p=\lfloor n^c\rfloor\in\mathbb P. \] We also prove pointwise almost-everywhere convergence of the corresponding multiple ergodic averages.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

For every $c>1$ sufficiently close to $1$, we prove that there is $\kappa>0$ such that every sufficiently large $A\subset[N]$ of density at least $(\log N)^{-\kappa}$ contains \[ x,\ x+(p-1),\ x+(p-1)^2,\ldots,\ x+(p-1)^k, \qquad p=\lfloor n^c\rfloor\in\mathbb P. \] We also prove pointwise almost-everywhere convergence of the corresponding multiple ergodic averages.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	For every $c>1$ sufficiently close to $1$, we prove that there is $\kappa>0$ such that every sufficiently large $A\subset[N]$ of density at least $(\log N)^{-\kappa}$ contains \[ x,\ x+(p-1),\ x+(p-1)^2,\ldots,\ x+(p-1)^k, \qquad p=\lfloor n^c\rfloor\in\mathbb P.	result 0.96	0.95	0.85	asserted 1.00
s1	\] We also prove pointwise almost-everywhere convergence of the corresponding multiple ergodic averages.	result 0.97	0.94	0.16	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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References

Works cited (25)

[21]

X. Shao and Y.-C. Sun,Linear equations in Piatetski–Shapiro primes, arXiv:2605.18676v1 (2026)

View on Pith Pith review arXiv 2026
[23]

J. Ter¨ av¨ ainen,Pointwise convergence of ergodic averages with M¨ obius weight, arXiv:2401.03174v2 (2024)

View on Pith Pith review arXiv 2024
[17]

H. Mousavi,Lean formalization of polynomial configurations and pointwise averages along Piatetski–Shapiro primes, GitHub repository link

View on Pith
[1]

Y. Akbal and A. M. G¨ ulo˘ glu, Waring–Goldbach problem with Piatetski–Shapiro primes,J. Th´ eor. Nombres Bordeaux30(2018), no. 2, 449–467

View on Pith 2018
[2]

A. Balog and J. B. Friedlander, A hybrid of theorems of Vinogradov and Piatetski–Shapiro, Pacific J. Math.156(1992), no. 1, 45–62

View on Pith 1992
[3]

V. Bergelson and A. Leibman,Polynomial extensions of van der Waerden ’s and Szemer´ edi’s theorems, J. Amer. Math. Soc.9(1996), no. 3, 725–753

View on Pith 1996
[4]

J. Bourgain,An approach to pointwise ergodic theorems, in Geometric Aspects of Functional Analysis (1986/87), Lecture Notes in Math.1317, Springer, Berlin, 1988, 204–223

View on Pith 1986
[5]

A. Burgin,S´ ark¨ ozy’s theorem for shifted primes with restricted digits, arXiv:2510.13076 (2025)

View on Pith Pith review arXiv 2025
[6]

J. Fornal and B. Krause,Double recurrence and almost sure convergence: primes and weighted theory, arXiv:2603.22203

View on Pith arXiv
[7]

B. Green,On Landau–Siegel zeros, Hexagon preprint 2610.00311v2 (2026), https://hexagonmath.org/2610.00311v2

View on Pith Pith review arXiv 2026
[8]

B. Krause, H. Mousavi, T. Tao and J. Ter¨ av¨ ainen,Pointwise convergence of bilinear polynomial averages over the primes, Ergodic Theory Dynam. Systems (2025), doi:10.1017/etds.2025.10202; arXiv:2409.10510

View on Pith arXiv 2025
[9]

B. Krause, H. Mousavi, T. Tao and J. Ter¨ av¨ ainen,Quantitative bounds for sets lacking polynomial progressions with shifted prime difference, arXiv:2608.19525v1 (2026)

View on Pith Pith review arXiv 2026
[10]

P. LaVictoire,UniversallyL 1-bad arithmetic sequences, J. Anal. Math.113(2011), 241–263

View on Pith 2011
[11]

H. Li and H. Pan,The Green–Tao theorem for Piatetski–Shapiro primes, J. Funct. Anal.291 (2026), no. 1, Article 111440, doi:10.1016/j.jfa.2026.111440

View on Pith doi:10.1016/j.jfa.2026.111440 2026
[12]

J. Maynard,Primes with restricted digits, Invent. Math.217(2019), no. 1, 127–218

View on Pith 2019
[13]

M. Mirek,Roth’s theorem in the Piatetski–Shapiro primes, Rev. Mat. Iberoam.31(2015), no. 2, 617–656

View on Pith 2015
[14]

M. Mirek,ℓ p(Z)-boundedness of discrete maximal functions along thin subsets of primes and pointwise ergodic theorems, arXiv:1401.1361

View on Pith Pith review arXiv
[15]

M. Mirek, R. Wan and J. Wright,Pointwise convergence of polynomial multiple ergodic averages along the primes, arXiv:2505.15549v2 (2026). 14

View on Pith Pith review arXiv 2026
[16]

H. Mousavi,Polynomial configurations and pointwise averages along Piatetski-Shapiro sequences, arXiv:2609.01847. 2026 Sep 1

View on Pith Pith review arXiv 2026
[18]

I. I. Piatetski–Shapiro,On the distribution of prime numbers in sequences of the form[f(n)], Mat. Sb. (N.S.)33(75)(1953), no. 3, 559–566 (in Russian)

View on Pith 1953
[19]

A. S´ ark¨ ozy,On difference sets of sequences of integers. I, Acta Math. Acad. Sci. Hungar.31 (1978), no. 1–2, 125–149

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[20]

X. Shao and M. Wang,Quantitative bounds in a popular polynomial Szemer´ edi theorem, Proc. Roy. Soc. Edinburgh Sect. A (2025), First View, 1–27, doi:10.1017/prm.2025.10103

View on Pith doi:10.1017/prm.2025.10103 2025
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T. Tao,A sketch of the proof of non-existence of Siegel zeroes, What’s new, 19 October 2026, https://terrytao.wordpress.com/2026/10/19/ a-sketch-of-the-proof-of-non-existence-of-siegel-zeroes/

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View on Pith 2012

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [21]

X. Shao and Y.-C. Sun,Linear equations in Piatetski–Shapiro primes, arXiv:2605.18676v1 (2026)

View on Pith Pith review arXiv 2026

[2] [23]

J. Ter¨ av¨ ainen,Pointwise convergence of ergodic averages with M¨ obius weight, arXiv:2401.03174v2 (2024)

View on Pith Pith review arXiv 2024

[3] [17]

H. Mousavi,Lean formalization of polynomial configurations and pointwise averages along Piatetski–Shapiro primes, GitHub repository link

View on Pith

[4] [1]

Y. Akbal and A. M. G¨ ulo˘ glu, Waring–Goldbach problem with Piatetski–Shapiro primes,J. Th´ eor. Nombres Bordeaux30(2018), no. 2, 449–467

View on Pith 2018

[5] [2]

A. Balog and J. B. Friedlander, A hybrid of theorems of Vinogradov and Piatetski–Shapiro, Pacific J. Math.156(1992), no. 1, 45–62

View on Pith 1992

[6] [3]

V. Bergelson and A. Leibman,Polynomial extensions of van der Waerden ’s and Szemer´ edi’s theorems, J. Amer. Math. Soc.9(1996), no. 3, 725–753

View on Pith 1996

[7] [4]

J. Bourgain,An approach to pointwise ergodic theorems, in Geometric Aspects of Functional Analysis (1986/87), Lecture Notes in Math.1317, Springer, Berlin, 1988, 204–223

View on Pith 1986

[8] [5]

A. Burgin,S´ ark¨ ozy’s theorem for shifted primes with restricted digits, arXiv:2510.13076 (2025)

View on Pith Pith review arXiv 2025

[9] [6]

J. Fornal and B. Krause,Double recurrence and almost sure convergence: primes and weighted theory, arXiv:2603.22203

View on Pith arXiv

[10] [7]

B. Green,On Landau–Siegel zeros, Hexagon preprint 2610.00311v2 (2026), https://hexagonmath.org/2610.00311v2

View on Pith Pith review arXiv 2026

[11] [8]

B. Krause, H. Mousavi, T. Tao and J. Ter¨ av¨ ainen,Pointwise convergence of bilinear polynomial averages over the primes, Ergodic Theory Dynam. Systems (2025), doi:10.1017/etds.2025.10202; arXiv:2409.10510

View on Pith arXiv 2025

[12] [9]

B. Krause, H. Mousavi, T. Tao and J. Ter¨ av¨ ainen,Quantitative bounds for sets lacking polynomial progressions with shifted prime difference, arXiv:2608.19525v1 (2026)

View on Pith Pith review arXiv 2026

[13] [10]

P. LaVictoire,UniversallyL 1-bad arithmetic sequences, J. Anal. Math.113(2011), 241–263

View on Pith 2011

[14] [11]

H. Li and H. Pan,The Green–Tao theorem for Piatetski–Shapiro primes, J. Funct. Anal.291 (2026), no. 1, Article 111440, doi:10.1016/j.jfa.2026.111440

View on Pith doi:10.1016/j.jfa.2026.111440 2026

[15] [12]

J. Maynard,Primes with restricted digits, Invent. Math.217(2019), no. 1, 127–218

View on Pith 2019

[16] [13]

M. Mirek,Roth’s theorem in the Piatetski–Shapiro primes, Rev. Mat. Iberoam.31(2015), no. 2, 617–656

View on Pith 2015

[17] [14]

M. Mirek,ℓ p(Z)-boundedness of discrete maximal functions along thin subsets of primes and pointwise ergodic theorems, arXiv:1401.1361

View on Pith Pith review arXiv

[18] [15]

M. Mirek, R. Wan and J. Wright,Pointwise convergence of polynomial multiple ergodic averages along the primes, arXiv:2505.15549v2 (2026). 14

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	For every \(c>1\) sufficiently close to \(1\), we prove that there is \(\kappa>0\) such that every sufficiently large \(A\subset[N]\) of density at least \((\log N)^{-\kappa}\) contains \[ x,\ x+(p-1),\ x+(p-1)^2,\ldots,\ x+(p-1)^k, \qquad p=\lfloor n^c\rfloor\in\mathbb P.	result 0.96	0.95	0.85	asserted 1.00
s1	\] We also prove pointwise almost-everywhere convergence of the corresponding multiple ergodic averages.	result 0.97	0.94	0.16	asserted 1.00