Alladi Ramakrishnan Hall

Diophantine Approximation with prime restriction

Dwaipayan Mazumder

CMI

The distribution of $\alpha p$ modulo one, where $p$ runs over the rational primes and $\alpha$ is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents $
u>0$ one can establish the infinitude of primes $p$ satisfying $||\alpha p||\le p^{-
u}$. In that context we see the method of Vaughan who showed $0<
u< 1/4$ in brief and then Harman's idea to improve the range to get $0<
u <3/10$.