Inhomogeneous approximation for systems of linear forms with primitivity constraints (Q6651997)

Diophantine approximation of \(m\) linear forms in \(n\) variables represented by a matrix \(A\) concerns the problem of determining whether there are infinitely many pairs \((p,q) \in \mathbb{Z}^m \times \mathbb{Z}^n\) such that \(qA-p \in B(0,\psi(\vert q \vert))\). Here, \(\psi: \mathbb{N} \rightarrow \mathbb{R}_{\ge 0}\) is a function, \(\vert q \vert\) denotes the maximum modulus among the coordinates of \(q\) and \(B(0,r)\) denotes the open ball of radius \(r\) centred at the origin. The metric theory is concerned with the Lebesgue measure and Hausdorff dimension of the set of \(A\) for which this happens infinitely often.\N\NIn the present paper, the authors consider the related problem of inhomogenous approximation is considered under additional restrictions on the approximating vectors \((p,q)\). The inhomogeneous approximation considered arises from letting the balls have centres different from the origin. The centres need not be the same, and the ball \(B(0,\psi(\vert q \vert))\) are replaced by generic balls \(B_{\vert q \vert}\), with centre depending only on \(\vert q\vert\). Furthermore, the integer vectors \((p,q)\) are required to have co-prime coordinates in blocks given by some fixed partition \(\pi\) of \(m+n\).\N\NThe authors obtain null-full laws for these sets for Lebesgue measure and zero-infinity laws for Hausdorff measures under various assumptions on the decay of the measure of the balls as \(\vert q \vert\) increases, assumptions on the dimensions \(m\) and \(n\), as well as assumptions on the partition \(\pi\). For example, if we assume that the Lebesgue measure of the balls \(B_{\vert q \vert}\) is non-increasing and that \(mn > 2\), then the set of \(A\) for which there are infinitly many solutions to \(qA-p \in B_{\vert q \vert}\) with the coordinates of \((p,q)\) co-prime inside all parts of the partition \(\pi\) of \(m+n\), where \(\pi\) consists of parts of cardinality at least \(2\), is full (resp. null) with respect to Lebesgue measure depending on the divergence (resp. convergence) of the series \(\sum q^{n-1} \vert B_q\vert\), where \(\vert B_q \vert\) denotes the Lebesgue measure of the ball \(B_q\). A number of results in this spirit are obtained; and the assumption that the measures of the balls satisfies a decay condition is also discussed and removed for certain partitions.