On a problem of Erdös and Szemerédi

Abstract

In 1966 P. Erdös proved the following theorem:

Let $\text{B}$ = {b_i: 1 < b₁ < b₂ < b₃ < …} be an infinite sequence of integers, such that

$$\text{(b}{}_{\mathrm{i}}\text{,b}{}_{\mathrm{j}}\text{=1}\text{for all}\text{i ≠ j}\text{and}\text{∑}\text{1=i}\text{∞}\text{1}\text{b}{}_{\mathrm{i}}\text{<∞}$$

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 $\text{B}$. Szemerédi showed that Erdös' theorem holds true for $\text{α =}\text{1}\text{2}\text{+ ϵ}$ and in our paper we prove, that $\text{1}\text{2}\text{+ ϵ}$ may be replaced by $\text{9}\text{20}\text{+ ϵ}$. The proof uses nontrivial estimates of exponential sums.