Zi-Rui Jiang

Metric Diophantine approximation with primes on irrational lines

Let $c>1$ be irrational. We prove that, for almost every $x>0$ and every $\varepsilon>0$, there are infinitely many triples $(p,r,s)$, with $p$ and $r$ prime and $s$ an integer, for which \[ 0<px-r<p^{-1/3+\varepsilon},\qquad 0<pcx-s<p^{-1/3+\varepsilon}. \] This improves the exponent $1/5$ in the homogeneous-line case...