• 1 Tata Institute for Fundamental Research

K. F. Roth proved in 1957 that if $1 = q_1 < q_2 <q_3\ldots$ is the sequence of square-free integers, then $q_{n+1} - q_n = O(n^{\frac{3}{13}}(\log n)^{\frac{4}{13}})$. P. Erdös set the problem in more general conditions. If $2 \leq b_1 < b_2 < b_3\ldots$ is a sequence of mutually coprime integers, and $\sum\frac{1}{b_i} < \infty$, he proved that if $1 = q_1 < q_2 <q_3\ldots$ is the sequence of integers not divisible by any $b_i$, then $$ q_{n+1} - q_n = O\left(q_n^{\theta}\right),$$ where $0<\theta<1.$ C. Szemeredi made an important progress and proved that if $Q(x)=\sum_{q_i\leq x} 1$, then $Q(x+h)-Q(x) >\!\!> h,$ where $h \geq x^{\theta}.$ Refining some of Szemeredi's ideas, it is proved in this paper that %if 0 < < 1, and $\sum\frac{1}{b_i}<\infty$, then $$Q(x+h) - Q(x) >\!\!> h,$$ where $x\geq h \geq x^{\theta}$ and $\theta >\frac{1}{2}$ is any constant. %In the later part, using the ideas of Jutila, Brun and I. M. Vinogradov, a stronger version (Theorem 2) is proved.