Diophantine approximation with Goldbach numbers

Abstract

In this paper we consider Diophantine Approximation using numbers which are a sum of two odd primes (Goldbach Numbers). We use exponential sums and sieve methods to establish, for any irrational $\alpha$ and arbitrary real $\beta$, that there are infinitely many solutions to \[ ||\alpha n + \beta|| < n^{- \frac56}, n = p_1+p_2, p_j \text{denotes an odd prime}, \] where $||\cdot||$ denotes, as usual, distance to a nearest integer.