An explicit power-saving bound for S\'{a}rk\"{o}zy's Theorem on shifted primes, assuming GRH

Minh Quang Nguy\^en; Qijun Gu

REVIEW

An explicit power-saving bound for S\'{a}rk\"{o}zy's Theorem on shifted primes, assuming GRH

Qijun Gu, Minh Quang Nguyên

Number Theory · Submitted 13 Sep 2026

Pith's verdict: The method is clean and the approach is different from Green's, but the claimed exponent 79/80 does not follow from the stated constraints — the optimization problem in Section 2 is inconsistent.

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arXiv ID2609.14471 (version 2)

Pith IDY55YL26V, Pith's permanent record of this paper and its review

CitationMinh Quang Nguy\^en, Qijun Gu. "An explicit power-saving bound for S\'{a}rk\"{o}zy's Theorem on shifted primes, assuming GRH." arXiv:2609.14471 (2026). https://pith.science/paper/Y55YL26V

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Editor's note

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The paper gives a conditional proof of a power-saving bound for Sarkozy's theorem on shifted primes, assuming GRH. The approach is genuinely different from Green's: it uses convolution with sieve weights (lambda_R, rho_T) instead of Ramanujan sums, and the major-arc analysis is worked out carefully. The organisation is clear and the technical machinery is laid out in a way that could be useful for other additive problems. I think the authors have done real work here.

But there is a load-bearing problem that the stress-test note got right. Proposition 2.1 gives ||bf - gw||_inf << N^{1+o(1)}(R^{-1/2}+R^{7/2}/N^{1/2}) Q T^3. With the optimised parameters Q = T = N^{1/48}, R = N^{1/8}, the first term becomes R^{-1/2} Q T^3 = N^{-1/16 + 1/48 + 3/48} = N^{1/48}, so the bound is N^{1+1/48+o(1}}, i.e. not a power saving at all. The optimisation problem in Section 2 gives delta = min(..., 1/2 y - x - 3w, ...) and claims delta = 1/80 from these parameters, but plugging in x = w = 1/48, y = 1/8 makes 1/2 y - x - 3w = 1/16 - 4/48 = -1/48, so the min is negative — the claimed exponent is unsupported. The error is not a typo: the constraint that should enforce power saving is x+3w < y/2 (or an equivalent inequality), and the given parameters violate it. This gap undermines the central quantitative claim.

Apart from this, the rest of the paper looks like a solid technical piece: the major-arc decomposition (Proposition 5.1) is heavy but appears derived correctly, and the error estimates are detailed. The reliance on GRH is clearly stated, and the choice to work with Lambda_R and tau_{4,T} is well motivated.

My take: the method is worth publishing eventually, but the current version has a mistake that makes the headline result unsubstantiated. A careful revision is needed – either a different set of parameters that actually satisfies the constraints, or a corrected optimisation. If the authors can fix this, the paper would be a nice contribution. As it stands, I would not cite it.

Recommendation: send to peer review, but the referee should be asked to verify the optimisation and the consistency of the constraints. The flaws are serious enough that I would not accept the paper in its present form, but the underlying work is serious enough to warrant a revision.

Referee report

Major objections (2)

[Section 2 (end) and optimisation] The optimisation problem and the claimed exponent δ=1/80 are inconsistent with the bounds derived in Propositions 2.1–2.4. With the proposed parameters T=Q=N^{1/48}, R=N^{1/8}, the error term in Proposition 2.1 is ∥bf−ĝw∥∞ ≪ N^{1+o(1)} (R^{-1/2}+R^{7/2}N^{-1/2}) Q T^3 = N^{1+o(1)} N^{-1/16+1/12} = N^{1+1/48+o(1)}, which is not a power saving (it is superlinear). Thus the condition for δ0>0 is violated, and the exponent 1/80 is unsupported. The paper must either correct the optimisation (possibly the signs in the constraints are wrong) or provide a valid parameter choice that yields a positive power saving. Without a fix, the central quantitative claim is unfounded.
[Section 6 (e.g., Proposition 6.3, Lemma 6.2, Proposition 6.4)] Several key calculations in Section 6 are left as 'exercises to the reader' or are omitted (e.g., the eight-term sum in Proposition 6.3, the bounding in Proposition 6.4, and the derivation of the product forms in Lemma 6.2). For a journal publication, these steps should be explicted or at least sketched convincingly; the reader cannot be expected to fill them in. Incomplete derivation undermines the verifiability of the main term and error bounds, which are crucial for Proposition 5.1.

Minor objections (4)

[Throughout] The paper contains many typos and formatting issues: e.g., 'S´ark¨ozy' in the title, inconsistent spacing in equations, and missing punctuation in some displayed equations. These should be corrected for readability.
[Section 3, Lemma 3.1] The lemma quotes Soundararajan's lemma; it should provide a precise citation (the lemma from [15] is Lemma 1, but the reference list says 2007 lecture notes, which may not be easily accessible; consider citing a more standard source or providing a short proof.)
[Section 5, Lemma 5.1] The error term O(Q^{-2} ||g||_1) is cited but not fully justified; the bounding of ||g||_1 should appear earlier or be referenced.
[Appendix A, Lemma A.2] The bound in Lemma A.2 is standard, but the proof is abbreviated; a short derivation or a reference to a text (e.g., Vaughan) would be helpful.

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The paper

The paper claims

Assuming GRH, a set avoiding shifted primes has size at most $N^{79/80+o(1)}$, an explicit power saving for Sárközy's theorem.

The reading

The paper aims to prove an explicit power-saving bound for Sárközy's theorem on shifted primes under the Generalised Riemann Hypothesis (GRH). It shows that if a subset $A$ of $\{1,\dots,N\}$ contains no two elements differing by $p+1$ for any prime $p$, then $|A| \ll N^{79/80+o(1)}$. This improves on Green's recent ineffective power-saving result by providing a concrete exponent, at the cost of assuming GRH. The result matters because it gives the first explicit quantitative bound for a classical problem in additive combinatorics, and the techniques—Fourier analysis combined with sieve weights—offer a transparent alternative to previous approaches. If correct, it also suggests that GRH can be used to obtain effective constants in related prime difference problems.

What carries the argument

The central object is the function $f(n) = \Lambda'(n-1)\tau_{4,T}(n)\Lambda_Q(n+1)w(n)$ and its Fourier transform, where $\Lambda'$ is the modified von Mangoldt function on primes, $\tau_{4,T}$ is a restricted divisor function, $\Lambda_Q$ is a Heath-Brown sieve weight, and $w(n)=1-n/N$ is a smooth cutoff. The key machinery is a Fourier-closeness lemma (Proposition 2.1) that approximates $\widehat{f}$ by $\widehat{gw}$ (with $\Lambda_R$ in place of $\Lambda'$) using GRH via Soundararajan's exponential sum bound; minor arc bounds for $\widehat{gw}$ via sieve decomposition and Vinogradov's estimate; and a major arc decomposition that expresses $\widehat{gw}$ in terms of short-interval sums $H_{q,a}(t)$, which are shown to be positive constants plus controllable errors (Proposition 5.1). The optimisation of the parameters $R,Q,T$ produces the exponent $1/80$.

Extended reading notes

Core claim

Theorem 1.1: Assume the Generalised Riemann Hypothesis. If $A\subseteq\{1,\dots,N\}$ contains no two elements differing by $p+1$ for any prime $p$, then $|A|\ll N^{79/80+o(1)}$. The proof uses the Fourier transform identity $0 = \int_0^1 |\widehat{1_A}|^2(\alpha)\widehat{f}(\alpha)\,d\alpha$ with a carefully constructed function $f$ involving the von Mangoldt function, a fourth divisor function, and a sieve weight. By decomposing the integral into major and minor arcs, bounding the minor-arc contribution via Vinogradov's estimate, and using GRH to control the major-arc Fourier closeness, the authors obtain a positive lower bound for the contribution near $\alpha=0$ and a negative upper bound away from zero, forcing $|A|\ll N^{1-\delta+o(1)}$. Optimising the parameters $T=Q=N^{1/48}$, $R=N^{1/8}$ yields $\delta=1/80-o(1)$.

Load-bearing premise

The proof assumes the Generalised Riemann Hypothesis (GRH) to obtain a sufficiently good bound on the Fourier transform of the von Mangoldt function in Lemma 3.1; without GRH this estimate fails and the argument does not go through.

Editorial extensions

If this is right

If true, this provides the first explicit power-saving exponent ($1/80$) for Sárközy's theorem on shifted primes under any hypothesis, making the bound concrete rather than ineffective.
The method demonstrates that combining Fourier analysis with sieve weights (such as the Heath-Brown $\Lambda_R$) can yield effective bounds in additive problems involving primes, and may be adapted to other shifted-prime configurations.
The exponent $1/80$ is not claimed to be optimal; the paper leaves an optimisation exercise that could be refined, potentially improving the saving with the same framework.
The conditional use of GRH suggests that unconditional approaches (e.g., using the Bombieri–Vinogradov theorem or sieve methods) could produce a weaker but unconditional power saving, as the authors hint at a future work.
For function fields $\mathbb{F}_q[t]$, where GRH is known, analogous results already exist; this paper provides a transparent alternative to Green's strategy and may be transferable to that setting.

Reading between the lines

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

The fact that an explicit exponent emerges from a simple optimisation problem (with just three power-law parameters) suggests that the $1/80$ bound may be improvable by more sophisticated weight choices or by treating the major-arc errors more carefully—possible follow-up work could target $1/40$ or better.
The reliance on GRH solely for Lemma 3.1 (a single exponential sum bound) indicates that if an unconditional analogue of that bound were available (perhaps via averaged forms of GRH or large-sieve estimates), the whole argument would become unconditional, likely at the cost of a worse exponent.
The paper's conceptual shift from Green's use of Ramanujan sums to a convolution/sieve perspective may open the door to similar explicit bounds for other problems where the difference set avoids a sparse set defined by primes.

Original abstract

Assume the Generalised Riemann Hypothesis. If $A\subseteq \{1,...,N\}$ contains no two elements differing by $p+1$, $p$ prime, then $|A|\ll N^{\frac{89}{90} + o(1)}$.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,298 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.68 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.06 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

Assume the Generalised Riemann Hypothesis. If $A\subseteq \{1,...,N\}$ contains no two elements differing by $p+1$, $p$ prime, then $|A|\ll N^{\frac{79}{80} + o(1)}$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Assume the Generalised Riemann Hypothesis.	background 0.54	0.12	0.02	unclear 0.61
s1	If $A\subseteq \{1,...,N\}$ contains no two elements differing by $p+1$, $p$ prime, then $\|A\|\ll N^{\frac{79}{80} + o(1)}$.	result 0.99	0.84	0.95	asserted 1.00

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

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

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figures and tables
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relations to other papers

levels run

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Discussion

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