K. F. Roth proved in 1957 that if 1=q1<q2<q3… is the sequence of square-free integers, then qn+1−qn=O(n313(logn)413). P. Erdös set the problem in more general conditions. If 2≤b1<b2<b3… is a sequence of mutually coprime integers, and ∑1bi<∞, he proved that if 1=q1<q2<q3… is the sequence of integers not divisible by any bi, thenqn+1−qn=O(qθn),
where 0<θ<1. C. Szemeredi made an important progress and proved that if Q(x)=∑qi≤x1, then Q(x+h)−Q(x)>>h, where h≥xθ. Refining some of Szemeredi's ideas, it is proved in this paper that %if 0 < < 1, and ∑1bi<∞, thenQ(x+h)−Q(x)>>h,
where x≥h≥xθ and θ>12 is any constant. %In the later part, using the ideas of Jutila, Brun and I. M. Vinogradov, a stronger version (Theorem 2) is proved.