On Harrap's conjecture in Diophantine approximation

TL;DR

This paper proves a conjecture by Harrap concerning inhomogeneous linear Diophantine approximation and properties of BAD sets, advancing understanding in this area of number theory.

Contribution

It provides a proof of Harrap's conjecture related to BAD sets in inhomogeneous Diophantine approximation, a previously unresolved problem.

Findings

Confirmed Harrap's conjecture on BAD sets

Enhanced understanding of inhomogeneous Diophantine approximation

Established new properties of BAD(,) sets

Abstract

We prove a conjecture due to Stephen Harrap on inhomogeneous linear Diophantine approximation related to ${\rm BAD}(\alpha,\beta)$ sets.