On Sárközy's theorem for shifted primes
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 \(a_i\in \mathbb{R}\) and which satisfy \(T(0)=1\), \(T(x)\ge 0\) for all \(x\). The van der Corput property is that \(\gamma(N)\to 0\) as \(N\to \infty\). This formulation of the property is due to \textit{T. Kamae} and \textit{M. Mendès France} [Isr. J. Math. 31, 335--342 (1978; Zbl 0396.10040)] and \textit{I. Z. Ruzsa} [Colloq. Math. Soc. János Bolyai 34, 1419--1443 (1984; Zbl 0572.10035)]. Kamae and Mendès France proved that the shifted primes have the van der Corput property, but did not provide explicit bounds. The bound \(\gamma(N)\ll (\log N)^{-1+o(1)}\) was subsequently proved by \textit{S. Slijepčević} [Funct. Approximatio, Comment. Math. 48, No. 1, 37--50 (2013; Zbl 1329.11076)].\N\NThe second main result of the article under review is the following upper bound for \(\gamma(N)\).\N\N\textit{Theorem \textsl{1.2}.} We have \(\gamma(N)\ll N^{-c}\) for some absolute constant \(c>0\). That is, there is a cosine polynomial \(T\in \mathcal{T}_N\) with \(a_0\ll N^{-c}\).\N\NInstead of directly working with the van der Corput property as defined above, the author derives bounds for the relative density \(\delta(N):=|A|/N\) of the set \(A\) and the quantity \(\gamma(N)\) from the following proposition, which captures a property that is equivalent to the van der Corput property.\N\N\noindent \textit{Proposition \textsl{2.1}.} Suppose that there is a function \(\Psi:\mathbb{Z}\rightarrow \mathbb{R}\) and \(\delta_1, \delta_2>0\) with the following properties:\N\N(1) \(\Psi(n)\) is supported on the set \(\{p-1:p\;\text{prime}\}\);\N\N(2) \(\sum_{n=1}^N \Psi(n)\cos(2\pi n\theta)\ge -\delta_1 N\), for all \(\theta \in \mathbb{R}/\mathbb{Z}\);\N\N(3) \(\sum_{n=1}^N \Psi(n)\ge \delta_2 N\).\N\N\noindent Then \(\delta(N)\le \frac{2\delta_1}{\delta_1+\delta_2}\) and \(\gamma(N)\le \frac{\delta_1}{\delta_1+\delta_2}\). Conversely, there are \(\delta_1, \delta_2>0\) and a function \(\Psi\) satisfying (1), (2) and (3) above and such that \(\gamma(N)= \frac{\delta_1}{\delta_1+\delta_2}\).\N\NAn immediate corollary is that \(\delta(N)\le 2\gamma(N)\).\N\NTheorem 1.1 and Theorem 1.2 are both consequences of the following result (which is equivalent to Theorem 1.2). \N\N\textit{Theorem \textsl{2.2}}. There are absolute constants \(0<\kappa_2<\kappa_1\) and a function \(\Psi:\mathbb{Z}\rightarrow \mathbb{R}\) with the following properties:\N\N(1) \(\Psi(n)\) is supported on the set \(\{p-1:p\;\text{prime}, \;p\le N\}\);\N\N(2) \(\sum_{n=1}^N \Psi(n)\cos(2\pi n\theta)\ge -N^{1-\kappa_1}\), for all \(\theta \in \mathbb{R}/\mathbb{Z}\);\N\N(3) \(\sum_{n=1}^N \Psi(n)\ge N^{1-\kappa_2}\).\N\NThe bulk of this 81-page article is concerned with the proper construction of the function \(\Psi\) and the proof that it satisfies the conclusions of Theorem 2.2.\N\NThe paper is wonderfully written, featuring clear and careful exposition and a thoughtful organization. After an introductory first section, the reader can find a sketch of the construction of the function \(\Psi\) in Proposition 2.1 and a helpful plan of the paper. Part II deals with the definitions of certain approximants \(\Lambda_\sharp\) to the von Mangoldt function \(\Lambda\), which are very close to it in Fourier space. Part III of the paper assembles various preliminaries for the main argument. In Part IV the function \(\Psi\) is rigorously constructed and shown to have the desired properties. Part V contains five appendices covering Postnikov character formula, various estimates for arithmetic functions, Dirichlet characters, an exponential sum estimate, and an explicit estimate for the \(\Gamma\)-function.\N\NThe author remarks that he has made a significant effort to ensure that the key intermediate results are self-contained wherever possible, allowing readers to grasp the basic structure of the argument without needing to read all the proofs. The paper contains numerous hyperlinks to help recall important definitions and intermediate results.
scientific article; zbMATH DE number 7887950
scientific article