For irrationals α with bounded continued fraction terms, the number of primes p in (X-Y, X] with ||pα|| < δ is asymptotically 2δ Y / log X when X^{2/3+ε} ≤ Y ≤ X/2 and δ satisfies the given lower bound.
Harman,On the distribution ofαpmodulo one, J
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Diophantine approximation with primes from short intervals
For irrationals α with bounded continued fraction terms, the number of primes p in (X-Y, X] with ||pα|| < δ is asymptotically 2δ Y / log X when X^{2/3+ε} ≤ Y ≤ X/2 and δ satisfies the given lower bound.