Diophantine approximation with primes from short intervals

In this paper, we establish hybrid results on Diophantine approximation with primes from short intervals. In particular, we prove the following result in a slightly modified form: If $\alpha$ is an irrational number having a continued fraction expansion with bounded terms (in particular, if $\alpha$ is a quadratic irrational), then the number of primes $p$ in the interval $(X-Y,X]$ satisfying $\mathrm{\mid }\mathrm{\mid }p\alpha \mathrm{\mid }\mathrm{\mid }<\delta$ is asymptotically equal to $2\delta Y\mathrm{/}\log X$, provided that $X\ge 10$, $X^{2\mathrm{/}3+\epsilon }\le Y\le X\mathrm{/}2$ and $X^{\epsilon }\max \{X^{1\mathrm{/}4}{Y}^{-1\mathrm{/}2},X^{2\mathrm{/}3}{Y}^{-1}\}\le \delta \le 1\mathrm{/}2$.