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The Hausdorff dimension of systems of simultaneously small linear forms. (English) Zbl 0793.11019

The Hausdorff dimension of \(\widehat{W} (m,n;\tau)\), the set of systems of real linear forms (identified with \(m\times n\) real matrices) for which there exist infinitely many \({\mathbf q}\in \mathbb{Z}^ m\) such that the inequalities \[ \biggl| \sum_{i=1}^ m q_ i x_{ij} \biggr|< |{\mathbf q}|^{-\tau}, \qquad j=1,\dots, n, \] where \(|{\mathbf q}|= \max \{| q_ 1|,\dots, | q_ n|\}\), hold is shown to be \((m-1) n+m/ (\tau+1)\) if \(\tau> (m/n)-1\) and \(mn\) otherwise. This question is connected with small denominator problems in dynamical systems.
As is usual, the main difficulty is establishing the downward inequality for the dimension. When \(m>n\), this is obtained by a straightforward application of ubiquity. This approach fails when \(m\leq n\) but in this case a subset of \(\widehat{W}(m,n;\tau)\) is then diffeomorphic to the Cartesian product of an \((m-1) (n-m+1)\)-dimensional cube and \(\widehat{W} (m,m-1;\tau)\); this gives the result.

MSC:

11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
11J83 Metric theory
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References:

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