On the distribution of fractions with power denominator

Abstract

In this paper we obtain a sharp upper bound for the number of solutions to a certain diophantine inequality involving fractions with power denominator. This problem is motivated by a conjecture of Zhao concerning the spacing of such fractions in short intervals and the large sieve for power modulus. As applications of our estimate we show Zhao's conjecture is true except for a set of small measure and give a new ${\ell }_1\to {\ell }_2$ large sieve inequality for power modulus.

arXiv commentThere was an error in applying the van der Corput transform in the previous version. Correcting this error led to a simplification of the argument

References in corpus (2)

• A new bound for the large sieve inequality with power moduli
• The large sieve with square norm moduli in Z[i]