The metrical theory of simultaneously small linear forms (Q351235)

Given \(m,n\in\mathbb{N}\), a positive monotonically decreasing function \(\psi\), let \(W_0(m,n,\psi)\) be the set \(X\in[-1/2,1/2]^{mn}\) (\(m\times n\) matrices with entries in \([-1/2,1/2]\) such that \(|\mathbf{q}X|<\psi(|\mathbf{q}|)\) holds for infinitely many \(\mathbf{q}\in\mathbb{Z}^m\), where \(|\cdot|\) is the supremum norm. In the paper under review the authors investigate the Lebesgue and Hausdorff measure of the sets \(W_0(m,n,\psi)\) depending on the properties of \(\psi\) and to some extant on \(m\) and \(n\). The main results are Khintchine and Jarník type theorems and generalise a result of \textit{H. Dickinson} [Mathematika 40, No. 2, 367--374 (1993; Zbl 0793.11019)] for the Hausdorff dimension. NEWLINENEWLINEIn the classical theory of Diophantine approximation one investigates the set \(W(m,n,\psi)\) which is defined via the inequality \(\|\mathbf{q}X\|<\psi(|\mathbf{q}|)\) with \(\|\cdot\|\) being the distance to the nearest integer point. The seemingly `technical' change from the norm \(\|\cdot\|\) on the torus \(\mathbb{R}^n/\mathbb{Z}^n\) to the norm \(|\cdot|\) on \(\mathbb R^n\) poses to a certain degree of challenge. As a result one has to distinguish two separate cases: \(m>n\) and \(m\leq n\). In the second case one has to restrict \(X\) to lie on a submanifold of \(m\times n\) matrices in order to obtain any meaningful results. In the case \(m>n\) the main result reads as follows:NEWLINENEWLINE\NEWLINENEWLINE\textbf{Theorem 1.} Let \(m>n\) and \(\psi:\mathbb N\to(0,+\infty)\) be a monotonic function. Let \(f:[0,+\infty)\to[0,+\infty)\) be a continuous function such that \(f(0)=0\), \(g(r):=r^{-(m-1)n}f(r)\) is increasing and \(r^{-mn}f(r)\) is monotonic. Let NEWLINE\[NEWLINE S(m,n,\psi):=\sum_{r=1}^\infty g\left(\frac{\psi(r)}{r}\right)r^{m-1}\,. NEWLINE\]NEWLINE Then, \(\mathcal{H}^f(W_0(m,n,\psi))\), the Hausdorff \(f\)-measure of \(W_0(m,n,\psi)\), is \(0\) whenever \(S(m,n,\psi)<\infty\), and it is equal to \(\mathcal{H}^f([-1/2,1/2]^{mn})\) whenever \(S(m,n,\psi)=\infty\).