Zero-one laws in simultaneous and multiplicative Diophantine approximation

TL;DR

This paper establishes zero-one laws in simultaneous and multiplicative Diophantine approximation, answering longstanding questions and employing advanced measure-theoretic techniques.

Contribution

It introduces new zero-one laws in Diophantine approximation, extending previous results with novel measure-theoretic methods and higher-dimensional analogues.

Findings

Established zero-one laws in both approximation settings

Answered two open questions of Beresnevich and Velani

Developed new measure-theoretic tools for Diophantine approximation

Abstract

Answering two questions of Beresnevich and Velani, we develop zero-one laws in both simultaneous and multiplicative Diophantine approximation. Our proofs rely on a Cassels-Gallagher type theorem as well as a higher-dimensional analogue of the cross fibering principle of Beresnevich, Haynes and Velani.