Exceptional sets for Diophantine inequalities

TL;DR

This paper uses a variant of the Davenport-Heilbronn method to analyze the size of the exceptional set of real numbers that are not well-approximated by values of a real diagonal form at integers, showing it has measure diminishing as N grows.

Contribution

It introduces a new application of Freeman's variant of the Davenport-Heilbronn method to bound the measure of exceptional sets for Diophantine inequalities.

Findings

Exceptional set in [-N,N] has measure O(N^{1-c})

Method effectively bounds the size of the exceptional set

Provides conditions under which the measure estimate holds

Abstract

We apply Freeman's variant of the Davenport-Heilbronn method to investigate the exceptional set of real numbers not close to some value of a given real diagonal form at an integral argument. Under appropriate conditions, we show that the exceptional set in the interval [-N,N] has measure O(N^{1-c}), for a positive number c.