Inhomogeneous approximation for systems of linear forms with primitivity constraints

TL;DR

This paper advances the understanding of inhomogeneous approximation for systems of linear forms under specific primitivity constraints, extending previous theorems and addressing longstanding conjectures in the field.

Contribution

It introduces new primitivity constraints for integer points in linear forms and proves a univariate inhomogeneous Duffin--Schaeffer type theorem for systems in at least three variables.

Findings

Strengthens a theorem of Dani, Laurent, and Nogueira.

Proves a univariate inhomogeneous Duffin--Schaeffer conjecture for systems of linear forms.

Addresses problems posed by Dani, Laurent, Nogueira, and Sprindžuk.

Abstract

We study (inhomogeneous) approximation for systems of linear forms using integer points which satisfy additional primitivity constraints. The first family of primitivity constraints we consider were introduced in 2015 by Dani, Laurent, and Nogueira, and are associated to partitions of the coordinate directions. Our results in this setting strengthen a theorem of Dani, Laurent, and Nogueira, and address problems posed by those same authors. The second primitivity constraints we consider are analogues of the coprimality required in the higher-dimensional Duffin--Schaeffer conjecture, posed by Sprind\v{z}uk in the 1970's and proved by Pollington and Vaughan in 1990. Here, with attention restricted to systems of linear forms in at least three variables, we prove a univariate inhomogeneous version of the Duffin--Schaeffer conjecture for systems of linear forms, the multivariate homogeneous version of which was stated by Beresnevich, Bernik, Dodson, and Velani in 2009 and recently proved by the second author.