On the distribution of fractions with power denominator

TL;DR

This paper establishes a sharp upper bound for solutions to a diophantine inequality involving fractions with power denominators, advancing understanding of their distribution and related sieve inequalities.

Contribution

It provides a new upper bound for solutions to a specific diophantine inequality and proves Zhao's conjecture for most cases, along with a novel large sieve inequality for power moduli.

Findings

Zhao's conjecture holds for all but a small measure set

Derived a sharp upper bound for solutions to the diophantine inequality

Introduced a new large sieve inequality for power modulus

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 $\ell_1 \rightarrow \ell_2$ large sieve inequality for power modulus.