• 1 Tata Institute for Fundamental Research

K. F. Roth proved in 1957 that if $1 = q_1 < q_2 \!\!> h,$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.