Restriction estimates with sifted integers

G. K. Viswanadham; Tanmoy Bera

arxiv: 2512.21640 · v3 · submitted 2025-12-25 · 🧮 math.NT

Restriction estimates with sifted integers

Tanmoy Bera , G. K. Viswanadham This is my paper

Pith reviewed 2026-05-16 19:36 UTC · model grok-4.3

classification 🧮 math.NT

keywords restriction estimatessifted integersGreen-Taoadditive combinatoricsFourier analysisnumber theorylocal conditions

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The pith

Restriction estimates hold for integers up to N sifted by arbitrary forbidden residues modulo small primes.

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

The paper establishes restriction estimates for the indicator function of integers up to N that have been sifted by local conditions at each prime p up to z. For a set P of primes and arbitrary subsets L_p of Z/pZ, the sifted set consists of those integers that avoid all residues in L_p modulo p. A sympathetic reader cares because these estimates control the Fourier transform of the sifted set in a way that supports applications in additive combinatorics. The work shows the estimates survive when the sifting sets L_p are chosen freely rather than fixed in advance.

Core claim

Let P be a subset of primes and for each prime p in P consider a subset L_p of Z/pZ. The authors provide restriction estimates with integers ≤ N sifted by (L_p) for p≤z, p in P. This generalizes a result of Green-Tao on the restriction estimates.

What carries the argument

The sifted set of integers ≤ N, formed by excluding residues belonging to each chosen L_p modulo p for p ≤ z in P.

If this is right

Restriction bounds now apply directly to sets defined by arbitrary local avoidance conditions at small primes.
The estimates support proofs about arithmetic progressions inside sifted sets whose local densities vary with the choice of L_p.
The result removes the need for fixed sifting sets and works for any collection of subsets L_p.
Applications become possible for sets that are sieved differently at different primes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The approach may combine with sieve methods to study patterns in almost-prime sets.
One could test whether the same estimates persist when the sifting is applied only to a thin subset of the primes rather than all p ≤ z.
The generalization opens the possibility of deriving density theorems for sets that are locally constrained in non-uniform ways.

Load-bearing premise

The collection of sifting primes P and the size of z relative to N must satisfy implicit uniformity conditions so that the sifted set remains dense enough for the Fourier bounds to apply.

What would settle it

A concrete counterexample in which the L^p norm of the Fourier transform of the indicator of a specific sifted set exceeds the claimed bound when z grows too quickly with N.

read the original abstract

Let $\mathcal{P}$ be a subset of primes and for each prime $p\in \mathcal{P}$, consider a subset $\mathcal{L}_p$ of $\mathbb{Z}/p\mathbb{Z}$. We provide restriction estimates with integers $\leq N$ sifted by $(\mathcal{L}_p)_{\substack{p\leq z\\ p\in \mathcal{P}}}$. This generalizes a result of Green-Tao [3] on the restriction estimates.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper generalizes Green-Tao restriction estimates to sifted sets with arbitrary local conditions L_p, but leaves the required ranges for z and the density of P unspecified.

read the letter

The paper generalizes Green-Tao restriction estimates to sifted integers where at each prime p up to z in a set P, we avoid an arbitrary subset L_p of residues. That's the core claim in the abstract. What stands out is the move to arbitrary local sifting conditions. Green-Tao had specific setups for their work on primes, and this tries to handle general L_p at each prime. If the proofs adapt the major-minor arc analysis without extra assumptions, that could open doors for more applications in sieve theory, like in Goldston-Pintz-Yildirim or other weighted sieves. The concern is whether the estimates require hidden bounds on the parameters. For the minor arcs to behave as in the original, the sifted density needs to stay reasonably large, which depends on how many primes are in P and how big z is compared to N. The abstract does not mention any such restrictions on z/N or the density of P, so it's not clear if the result holds for all choices or only when the product of (1 - |L_p|/p) is not too small. The stress test highlights this as a potential issue for the minor-arc bounds. If the full paper includes explicit conditions or shows the estimates survive when the density drops a bit, that would strengthen the work. Without that, the generalization might only apply in the regime where the sifting is mild enough not to disrupt the exponential sum estimates. This is aimed at number theorists working on additive problems with sieves. A reader who knows the Green-Tao paper would see the extension quickly and could check if it applies to their own settings. It looks like it deserves peer review to check the details and see if the claims are fully supported by the arguments.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to establish restriction estimates for the sifted set A = {n ≤ N : n mod p ∉ L_p for all p ≤ z with p ∈ P}, where P is a subset of primes and each L_p is an arbitrary subset of Z/pZ. This is presented as a direct generalization of the restriction estimates obtained by Green-Tao.

Significance. If the estimates hold with explicit parameter ranges, the result would extend the scope of restriction theory to sifted sets, which appear in applications to almost-primes and sieve-theoretic problems in additive combinatorics. The allowance for arbitrary L_p strengthens the statement relative to the classical Green-Tao setting, provided the minor-arc analysis closes uniformly.

major comments (1)

[Abstract] Abstract and §1: the claimed generalization is stated for arbitrary L_p without recording the necessary constraints on z/N and the density of P (e.g., that the product δ = ∏ (1 - |L_p|/p) ≫ N^{-c} or that z remains small enough for the exponential-sum estimates on minor arcs to survive the sieve weights). These bounds are load-bearing for the major/minor-arc decomposition used in restriction estimates and must be stated explicitly for the scope of the theorem to be verifiable.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the suggestion to clarify the parameter ranges. We agree that the abstract and opening of §1 should explicitly record the constraints on z and the sifted density δ that are already present in the main theorems.

read point-by-point responses

Referee: [Abstract] Abstract and §1: the claimed generalization is stated for arbitrary L_p without recording the necessary constraints on z/N and the density of P (e.g., that the product δ = ∏ (1 - |L_p|/p) ≫ N^{-c} or that z remains small enough for the exponential-sum estimates on minor arcs to survive the sieve weights). These bounds are load-bearing for the major/minor-arc decomposition used in restriction estimates and must be stated explicitly for the scope of the theorem to be verifiable.

Authors: We agree that the abstract and the first paragraphs of §1 do not list the constraints explicitly. The main result (Theorem 1.2) already requires that z ≤ N^θ for a sufficiently small absolute θ > 0 and that the sifting density satisfies δ ≫ N^{-c} for a small absolute c > 0; these ensure that the minor-arc exponential-sum bounds survive the introduction of the sieve weights and that the major-arc analysis remains uniform. We will revise the abstract to read: “We establish restriction estimates for the sifted set A = {n ≤ N : n mod p ∉ L_p for all p ≤ z, p ∈ P}, provided z ≤ N^θ and δ ≫ N^{-c} for small absolute constants θ, c > 0.” We will also insert a short paragraph at the end of the introduction summarizing these ranges and referring the reader to the precise statements in §2. This change will be incorporated in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity: direct generalization of external Green-Tao result with no self-referential reduction

full rationale

The paper states it provides restriction estimates for integers sifted by arbitrary (L_p) and explicitly frames this as a generalization of Green-Tao [3]. No equations, definitions, or steps in the provided abstract reduce a claimed prediction or uniqueness result to a fitted parameter or self-citation by construction. The cited Green-Tao work is external (distinct authors), and the abstract records no ansatz smuggling, renaming of known patterns, or load-bearing self-citation. The derivation chain therefore remains independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities are mentioned or detailed.

pith-pipeline@v0.9.0 · 5356 in / 980 out tokens · 34244 ms · 2026-05-16T19:36:05.293206+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We provide restriction estimates with integers ≤ N sifted by (L_p) for p≤z, p in P. This generalizes a result of Green-Tao [3] on the restriction estimates.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.3 ... dual large sieve type inequality ... enveloping sieve of Ramaré and Ruzsa

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[1]

Bourgain

J. Bourgain. On Λ(p)-subsets of squares.Israel Journal of Mathematics67(1989), 291-311

work page 1989
[2]

B. Green. Roth’s theorem in the primes.Ann. Math.161(2005), 1609-1636

work page 2005
[3]

Green and T

B. Green and T. Tao. Restriction Theory of the Selberg Sieve, with applications.Journal de Th´ eorie des Nombres de Bordeaux18(2006), 147-182

work page 2006
[4]

A. J. Harper. Minor arcs, mean values, and restriction theory for exponential sums over smooth numbers. Comp. Math.152(2016), 1121-1158

work page 2016
[5]

H. L. Montgomery. The analytic principle of the large sieve.Bull. Amer. Math. Soc.84(1978), 547-567

work page 1978
[6]

Kyle Petersen.Eulerian Numbers.Birkhauser Adv

T. Kyle Petersen.Eulerian Numbers.Birkhauser Adv. Texts, Birkhauser/Springer, New York, 2015

work page 2015
[7]

Ramar´ e

O. Ramar´ e. On Snirelman’s constant. Ann.Scuola Normale Sup. Pisa. Classe di Scienze21(1995), 645-706

work page 1995
[8]

Ramar´ e

O. Ramar´ e. Notes on restriction theory in the primes.Isr. J. Math.261(2024), 717–738. 14 TANMOY BERA AND G. K. VISW ANADHAM

work page 2024
[9]

Ramar´ e

O. Ramar´ e. Cusps of primes in dense subsequences-by passing theW-trick.Preprint. https://arxiv.org/abs/2412.11527

work page arXiv
[10]

Ramar´ e and I.Z

O. Ramar´ e and I.Z. Ruzsa. Additive properties of dense subsets of sifted sequences.Journal de Th´ eorie des Nombres de Bordeux131(2001), 559-581

work page 2001
[11]

Ramar´ e and G.K

O. Ramar´ e and G.K. Viswanadham, The trigonometric polynomial on sums of two squares, and additive problem and generalisation.Trans. Amer. Math. Soc., Accepted for publication. 1Harish-Chandra Research Institute, A CI of Homi Bhabha National Institute, Chhatnag Road, Jhunsi, Prayagraj, India 211019 2Department of Mathematical Sciences, IISER Berhampur, G...

work page

[1] [1]

Bourgain

J. Bourgain. On Λ(p)-subsets of squares.Israel Journal of Mathematics67(1989), 291-311

work page 1989

[2] [2]

B. Green. Roth’s theorem in the primes.Ann. Math.161(2005), 1609-1636

work page 2005

[3] [3]

Green and T

B. Green and T. Tao. Restriction Theory of the Selberg Sieve, with applications.Journal de Th´ eorie des Nombres de Bordeaux18(2006), 147-182

work page 2006

[4] [4]

A. J. Harper. Minor arcs, mean values, and restriction theory for exponential sums over smooth numbers. Comp. Math.152(2016), 1121-1158

work page 2016

[5] [5]

H. L. Montgomery. The analytic principle of the large sieve.Bull. Amer. Math. Soc.84(1978), 547-567

work page 1978

[6] [6]

Kyle Petersen.Eulerian Numbers.Birkhauser Adv

T. Kyle Petersen.Eulerian Numbers.Birkhauser Adv. Texts, Birkhauser/Springer, New York, 2015

work page 2015

[7] [7]

Ramar´ e

O. Ramar´ e. On Snirelman’s constant. Ann.Scuola Normale Sup. Pisa. Classe di Scienze21(1995), 645-706

work page 1995

[8] [8]

Ramar´ e

O. Ramar´ e. Notes on restriction theory in the primes.Isr. J. Math.261(2024), 717–738. 14 TANMOY BERA AND G. K. VISW ANADHAM

work page 2024

[9] [9]

Ramar´ e

O. Ramar´ e. Cusps of primes in dense subsequences-by passing theW-trick.Preprint. https://arxiv.org/abs/2412.11527

work page arXiv

[10] [10]

Ramar´ e and I.Z

O. Ramar´ e and I.Z. Ruzsa. Additive properties of dense subsets of sifted sequences.Journal de Th´ eorie des Nombres de Bordeux131(2001), 559-581

work page 2001

[11] [11]

Ramar´ e and G.K

O. Ramar´ e and G.K. Viswanadham, The trigonometric polynomial on sums of two squares, and additive problem and generalisation.Trans. Amer. Math. Soc., Accepted for publication. 1Harish-Chandra Research Institute, A CI of Homi Bhabha National Institute, Chhatnag Road, Jhunsi, Prayagraj, India 211019 2Department of Mathematical Sciences, IISER Berhampur, G...

work page