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Metrical theorems on systems of affine forms. (English) Zbl 1464.11074

One can begin with authors’ description of the present research:
“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 \(\mathbf q\mapsto \mathbf q X+\mathbf\alpha\), where \(\mathbf q\in\mathbb Z^m\) (viewed as a row vector), X is an \(m\times n\) real matrix and \(\mathbf\alpha\in \mathbb R^n\). The classical setting refers to the {dist}\((\mathbf q X+\mathbf\alpha, \mathbb Z^m)\) to measure the closeness of the integer values of the system \((X, \mathbf\alpha)\) to integers. The absolute value setting is obtained by replacing {dist}\((\mathbf q X+\mathbf \alpha, \mathbb Z^m)\) with {dist}\((\mathbf q X+\mathbf \alpha, \mathbf 0)\); and the more general mixed settings are obtained by replacing {dist}\((\mathbf q X+\mathbf\alpha, \mathbb Z^m)\) with {dist}\((\mathbf q X+\mathbf\alpha, \Lambda)\), where \(\Lambda\) is a subgroup of \(\mathbb Z^m\). 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 [D. Kleinbock, J. Number Theory 79, No. 1, 83–102 (1999; Zbl 0937.11030)].”
Certain notions and auxiliary known results are discussed. The special attention is given to the following notions and areas of investigations: an approximation function, the singly metric theory, the doubly metric theory, the set of inhomogeneous badly approximable affine forms, the hyperplane game, the absolute affine linear forms, etc. The attention is also given to different versions of the Khintchine-Groshev theorem and to Kleinbock’s conjecture. The importance of the Khintchine-Groshev theorem is noted. In addition, the basic notions of the Hausdorff measure and dimension are recalled. One can note that the slicing lemma and the mass transference principle are described. The main results and several auxiliary statements are proven with explanations.

MSC:

11J83 Metric theory
11J20 Inhomogeneous linear forms

Citations:

Zbl 0937.11030
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Full Text: DOI arXiv

References:

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