On Sárközy's theorem for shifted primes
type: Item
instance of: scholarly article
title: On Sárközy's theorem for shifted primes
author: Ben Green
published in: Journal of the American Mathematical Society
publication date: 2024-07-29T00:00:00Z
review text: Let \(N\) be a positive integer, and \(A\subset \{1,2,\ldots,N\}\) be a set such that the difference of any two elements is never one less than a prime number. The first explicit bound of the cardinality \(|A|\) of \(A\) was obtained in 1978 by \textit{A. Sárközy} [Acta Math. Acad. Sci. Hung. 31, 355--386 (1978; Zbl 0387.10034)]. He proved that \(|A|\ll N (\log\log N)^{-2-o(1)}\). Subsequently, \textit{I. Z. Ruzsa} and \textit{T. Sanders} [Acta Arith. 131, No. 3, 281--301 (2008; Zbl 1170.11023)] showed that \(|A|\ll Ne^{-c(\log N)^{1/4}}\) for some absolute constant \(c>0\). In 2020, the exponent \(1/4\) was improved to \(1/3\) by \textit{R. Wang} [J. Number Theory 211, 220--234 (2020; Zbl 1444.11024)].\N\NIt is worth noting that, even assuming the Generalized Riemann Hypothesis (GRH), the methods from the aforementioned works would only get an estimate of the shape \(|A|\ll Ne^{-c(\log N)^{1/2}}\). In the paper under review, the author obtains a sharper upper bound for \(|A|\), without any unproven hypothesis. \N\N\textit{Theorem \textsl{1.1}.} Let \(A\subset \{1,2,\ldots,N\}\) be a set such that \(A - A\) contains no number of the form \(p-1\), \(p\) a prime. Then \(|A|\ll N^{1-c}\) for some \(c>0\).\N\NA specific value of the constant \(c\) is not provided, but \textit{J. Thorner} and \textit{A. Zaman} [Forum Math. 36, No. 4, 1059--1080 (2024; Zbl 07896914)] have shown that \(c=10^{-18}\) is admissible. The author remarks that assuming GRH, it is possible to prove Theorem 1.1 with any \(c<1/12\).\N\NThe main novelty in the proof of Theorem 1.1 is that instead of following the density increment strategy common to the works cited above, the author focuses on studying the so-called \textit{van der Corput} property for the shifted primes. For each positive integer \(N\), define \N\[\N\gamma(N):=\inf_{T\in \mathcal{T}_N} a_0, \N\]\Nwhere \(\mathcal{T}_N\) denotes the set of all cosine polynomials \N\[\NT(x)=a_0+\sum_{p\le N} a_{p-1} \cos(2\pi(p-1)x),\N\]\Nwhere \(a_i\…
reviewed by: T. P. Peneva
Mathematics Subject Classification ID: 11P32
zbMATH DE Number: 7887950
zbMATH Keywords: difference sets
zbMATH Keywords: shifted primes
zbMATH Keywords: Sárközy's theorem
zbMATH Keywords: Fourier transform
zbMATH Keywords: exponential sums
zbMATH Keywords: zero-density theorems.
MaRDI profile type: Publication
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DOI: 10.1090/JAMS/1036
description: scientific article; zbMATH DE number 7887950