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 [Formula: see text] is an irrational number having a continued fraction expansion with bounded terms (in particular, if [Formula: see text] is a quadratic irrational), then the number of primes [Formula: see text] in the interval [Formula: see text] satisfying [Formula: see text] is asymptotically equal to [Formula: