Metric Diophantine approximation with primes on irrational lines

Source abstract

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 of a theorem of Baier and Ghosh. The arithmetic input is the quantitative prime-counting lower bound in Matomäki's proof concerning primes in shrinking neighborhoods of an irrational rotation. We obtain a local first-moment lower bound by fixing the prime numerator and applying the prime number theorem to the admissible prime denominators. Two Selberg sieve squares, together with explicit trigonometric interval approximations, give a pointwise upper bound whose error tends to zero in $L^1$ after normalization. A truncation and density-point argument then yields the almost-everywhere conclusion.