On a question of Sárközy on gaps of product sequences (Q607859)

Let \(A=\{a_1<a_2<\cdots\}\) be a set of positive integers. Define its product set \[ A\cdot A=\{a_ia_j;a_i\in A,a_j\in A\} =\{b_1<b_2<\cdots\}. \] \textit{A. Sárközy} [``Unsolved problems in number theory.'' Period. Math. Hung. 42, No. 1-2, 17--35 (2001; Zbl 1062.11002)] asked the following question: ``Suppose that the density of \(A\) is greater than \(\alpha>0.\) Is there a positive constant \(c=c(\alpha)\) such that \(b_{n+1}-b_n\leq c(\alpha)\) holds for infinitely many \(n\)?'' The question was answered in the affirmative in papers by \textit{G. Bérczi} [Period. Math. Hung. 44, No.2, 137--145 (2002; Zbl 1026.11017)] and \textit{C. Sándor} [Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 48, 3--7 (2005; Zbl 1121.11011)]. Bérczi took as hypothesis \[ {\underline d}A=\liminf_{n\rightarrow\infty}{{|A\cap\{1,2,\dots,N\}|}\over N}>\alpha>0 \] and Sándor \[ {\overline d}A=\limsup_{n\rightarrow\infty}{{|A\cap\{1,2,\dots,N\}|} \over N}>\alpha>0. \] In both aforementioned papers the dependence of \(c(\alpha)\) on the exponent of the bound \(\alpha\) of the density is specified. In the paper under review, the authors take as hypothesis that the upper Banach density \[ d^*A=\lim_{k\rightarrow\infty}\limsup_{n\rightarrow\infty}{{|A\cap\{n+1,n+2,\dots,n+k\}|}\over k} \] is strictly positive. Notice that for a given set \(A\) of positive integers, \( d^*A\geq {\overline d}A\geq {\underline d}A.\) The authors prove the following theorems: Theorem 1. For every \(\alpha\in]0,1[\) and every set \(A\) of positive integers with \(d^*A>\alpha,\) we have \(b_{n+1}-b_n\ll\alpha^{-3}\) infinitely often. Theorem 2. For every \(\alpha\in]0,1[,\) there exists a set \(A\) of positive integers with \(d^*A>\alpha\) and such that \(b_{n+1}-b_n\gg\alpha^{-3}\) for every \(n.\) They also prove analogous results for the difference \(b_{n+t}-b_n\) where \(t\) is a fixed integer, \(t\geq 2.\) It is striking that in this case, in the analogs of Theorems 1 and 2, \(\alpha^{-3}\) is replaced by \(t^2\alpha^{-4},\) that is, the best exponent of \(\alpha^{-1}\) is different when \(t=1\) and when \(t\geq2.\) In the proof, a theorem by \textit{P. Erdős} and \textit{P. Turán} [J. Lond. Math. Soc. 16, 212--215 (1941; Zbl 0061.07301, JFM 67.0984.03)] on Sidon sets is used. Reviewer's remark: If \({\underline d}A>\alpha>0\), the existence of a \(\displaystyle c(\alpha)={C\over{\alpha}}>0\) seems trivial; \(C\) depends on \(\min A.\) But both Sárközy's question and Bérczi's results ask/answer for the best exponent of \(\alpha^{-1}\) in \(c(\alpha);\) and this independently of \(\min A.\) Also products of \(k\) terms of \(A\) were looked at.