Diophantine approximation with prime denominators in short intervals

For any real number x, let $\|x\|$ denote the distance from x to the nearest integer, and let α be any irrational number. We prove that, for Y=X^γ with 7/12<γ≤ 1 and 0<θ<min \(12γ-7)/24, 1/9\, there exist infinitely many X for which there are primes p satisfying \|α p\|<p-θ, X-Y<p<X. Moreover, we obtain an asymptotic formula for the number of such primes. Our result extends the admissible range of short intervals in the corresponding result of Baier and Roy from 2/3<γ≤ 1 to 7/12<γ≤ 1, reaching Huxley's classical range for primes in short intervals and also improves the exponent θ in 2/3≤ γ≤ 3/4.