Metric bootstraps for limsup sets

TL;DR

This paper introduces bootstrap techniques in metric Diophantine approximation that allow transferring measure results from Euclidean spaces to systems of linear forms, simplifying proofs of various theorems.

Contribution

It develops a method to bootstrap measure results from Euclidean spaces to systems of linear forms, enabling easier proofs of related theorems in Diophantine approximation.

Findings

Bootstrap method successfully transfers measure results.

Simplifies proofs of existing theorems.

Applies to both positive and full measure cases.

Abstract

In metric Diophantine approximation, one frequently encounters the problem of showing that a limsup set has positive or full measure. Often it is a set of points in $m$-dimensional Euclidean space, or a set of $n$-by-$m$ systems of linear forms, satisfying some approximation condition infinitely often. The main results of this paper are bootstraps: if one can establish positive measure for such a limsup set in $m$-dimensional Euclidean space, then one can establish positive or full measure for an associated limsup set in the setting of $n$-by-$m$ systems of linear forms. Consequently, a class of $m$-dimensional results in Diophantine approximation can be bootstrapped to corresponding $n$-by-$m$-dimensional results. This leads to short proofs of existing, new, and hypothetical theorems for limsup sets that arise in the theory of systems of linear forms. We present several of these.