Metrical theorems on systems of affine forms

TL;DR

This paper advances metric number theory for affine forms, establishing Khintchine--Groshev and Jarník theorems in mixed and classical settings, and shows badly approximable forms are hyperplane winning, answering a question by Kleinbock.

Contribution

It proves new metric theorems for affine forms in mixed and classical settings and demonstrates that badly approximable forms are hyperplane winning.

Findings

Proved Khintchine--Groshev and Jarník theorems for mixed affine forms.

Established Jarník theorem for classical affine forms.

Showed badly approximable forms are hyperplane winning.

Abstract

In this paper we discuss metric theory associated with the affine (inhomogeneous) linear forms in the so called doubly metric settings within the classical and the mixed setups. We consider the system of affine forms given by $\qq\mapsto \qq X+\bfalpha$, where $\qq\in\Z^m$ (viewed as a row vector), $X$ is an $m\times n$ real matrix and $\bfalpha\in \R^n$. The classical setting refers to the ${\rm dist}(\qq X+\bfalpha, \Z^m)$ to measure the closeness of the integer values of the system $(X, \bfalpha)$ to integers. The absolute value setting is obtained by replacing ${\rm dist}(\qq X+\bfalpha, \Z^m)$ with ${\rm dist}(\qq X+\bfalpha, \0)$; and the more general mixed settings are obtained by replacing ${\rm dist}(\qq X+\bfalpha, \Z^m)$ with ${\rm dist}(\qq X+\bfalpha, \Lambda)$, where $\Lambda$ is a subgroup of $\Z^m$. We prove the Khintchine--Groshev and Jarn\'ik type theorems for the mixed affine forms and Jarn\'ik type theorem for the classical affine forms. We further prove that the sets of badly approximable affine forms, in both the classical and mixed settings, are hyperplane winning. The latter result, for the classical setting, answers a question raised by Kleinbock (1999).