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- W2893772001 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 $qqmapsto qq X+bfalpha$, where $qqinZ^m$ (viewed as a row vector), $X$ is an $mtimes n$ real matrix and $bfalphain R^n$. 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 Jarnik type theorems for the mixed affine forms and Jarnik 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)." @default.
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- W2893772001 date "2014-06-16" @default.
- W2893772001 modified "2023-09-24" @default.
- W2893772001 title "Metrical theorems on systems of small Affine forms" @default.
- W2893772001 hasPublicationYear "2014" @default.
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