Metrical theorems on systems of affine forms

Mumtaz Hussain*, Simon Kristensen, David Simmons

*Corresponding author for this work

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaperJournal articleResearchpeer-review

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 q↦qX+α, where q∈Zm (viewed as a row vector), X is an m×n real matrix and α∈Rn. The classical setting refers to the dist(qX+α,Zm) to measure the closeness of the integer values of the system (X,α) to integers. The absolute value setting is obtained by replacing dist(qX+α,Zm) with dist(qX+α,0); and the more general mixed settings are obtained by replacing dist(qX+α,Zm) with dist(qX+α,Λ), where Λ is a subgroup of Zm. We prove the Khintchine–Groshev and Jarník type theorems for the mixed affine forms and Jarník 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).

Original languageEnglish
JournalJournal of Number Theory
Volume213
Pages (from-to)67-100
Number of pages34
ISSN0022-314X
DOIs
Publication statusPublished - Aug 2020

Keywords

  • Hausdorff dimension
  • Hausdorff measure
  • Hyperplane winning
  • Jarnik theorem
  • Khintchine–Groshev theorem
  • Schmidt game

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