On the distribution of fractions with power denominator

Bryce Kerr

On the distribution of fractions with power denominator

Number Theory 2019-04-22 v2

Authors: Bryce Kerr

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.

Keywords

diophantine inequality power denominator fractions distribution large sieve power modulus zhao conjecture

Cite

@article{arxiv.1901.01687,
  title  = {On the distribution of fractions with power denominator},
  author = {Bryce Kerr},
  journal= {arXiv preprint arXiv:1901.01687},
  year   = {2019}
}

Comments

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

On the distribution of fractions with power denominator

Abstract

Keywords

Cite

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