On a problem of Erdös and Szemerédi

Abstract

In 1966 P. Erdös proved the following theorem: Let B = { b i : 1 < b 1 < b 2 < b 3 < …} be an infinite sequence of integers, such that (b i,b j=1 for all i ≠ j and ∑ 1=i ∞ 1 b i <∞ Then there exists a constant 0 < α < 1 with the following property: For sufficiently large x the interval ( x − x α , x] contains integers, that are divisible by no element of B . Szemerédi showed that Erdös' theorem holds true for α = 1 2 + ϵ and in our paper we prove, that 1 2 + ϵ may be replaced by 9 20 + ϵ . The proof uses nontrivial estimates of exponential sums.