On a problem of Erdős and Szemerédi
\textit{P. Erdős} proved [Acta Arith. 12, 175-182 (1966; Zbl 0147.02601)] the following: Let \({\mathcal B}=\{b_ i\}^{\infty}_{i=1}\) be a sequence of distinct positive integers such that \((b_ i,b_ j)=1\) for \(i\neq j\) and the sum of \(1/b_ i\) converges. Then there exists \(\alpha\) such that \(0<\alpha <1\) and for \(x\geq x_ 0\) the interval \((x-x^{\alpha},x]\) contains integers divisible by no element of \textit{\({\mathcal B}\). E. Szemerédi} [ibid. 23, 359-361 (1973; Zbl 0266.10045)] showed that one can take \(\alpha =+\epsilon\). Now the authors make further progress in this interesting problem by showing that \(\alpha =9/20+\epsilon\) is permissible, estimating nontrivially certain exponential sums by using ideas due mainly to H. Iwaniec.
scientific article; zbMATH DE number 3924917