Diophantine approximation with prime restriction in real quadratic number fields

The distribution of $$\alpha p$$ modulo one, where p runs over the rational primes and $$ is a fixed irrational real, has received lot attention. It natural to ask for which exponents $$\nu >0$$ one can establish infinitude satisfying $$||\alpha p||\le p^{-\nu }$$ . latest record in this regard Kaisa Matomäki’s landmark result =1/3-\varepsilon presents limit currently known technology. Recently, Glyn Harman, and, jointly, Marc Technau first-named author, investigated same problem context imaginary quadratic fields. Harman obtained an analog $$\mathbb {Q}(i)$$ his {Q}$$ , yields exponent =7/22$$ author produced analogue Bob Vaughan’s =1/4-\varepsilon all number fields class 1. In present article, we last-mentioned real 1 under certain Diophantine restriction. This setting involves additional complication infinite group units ring integers. Moreover, although basic sieve approach remains (we use ideal version Harman’s sieve), takes different flavor since it becomes truly 2-dimensional. We reduce eventually counting is, interestingly, related roots congruences. To approximate them, by Christopher Hooley based on theory binary forms.