On the distribution of αp modulo one in quadratic number fields

We investigate the distribution of αp modulo one in quadratic
number fields K with class number one, where p is restricted to prime elements
in the ring of integers of K. Here we improve the relevant exponent 1/4 obtained
by the first- and third-named authors for imaginary quadratic number fields [On
the distribution of αp modulo one in imaginary quadratic number fields with class number one, J. Th ́eor. Nombres Bordx. 32 (2020), no. 3, 719–760]) and by the first- and second-named authors for real quadratic number fields [Diophantine approximation with prime restriction in real quadratic number fields, Math. Z. (2021)] to 7/22. This generalizes a result of Harman [Diophantine approximation with Gaussian primes, Q. J. Math. 70 (2019), no. 4, 1505–1519] who obtained the same exponent 7/22 for Q(i) by extending his method which gave this exponent for Q [On the distribution of αp modulo one. II, Proc. London Math. Soc. 72, (1996), no. 3, 241–260]. Our proof is based on an extension of Harman’s sieve method to arbitrary number fields. Moreover, we need an asymptotic evaluation of certain smooth sums over prime ideals appearing in the above-mentioned work by the first- and second-named authors, for which we use analytic properties of Hecke L-functions with Gr ̈oßencharacters.