Uniform dilations in higher dimensions (Q2869837)

By a theorem of \textit{S. Glasner} [Isr. J. Math. 32, 161--172 (1979; Zbl 0406.54023)] for any infinite subset \(X\) of the torus \(\mathbb T=\mathbb R/\mathbb Z\) and \(\varepsilon>0\) there is a \(n\in\mathbb N\) such that the set \(nX\) is \(\varepsilon\)-dense in \(\mathbb T\). \textit{N. Alon} and \textit{Y. Peres} [Geom. Funct. Anal. 2, No. 1, 1--28 (1992; Zbl 0756.11020)] gave estimates on the number \(k(\epsilon)\) for which every set \(X\) of cardinality \(k(\varepsilon)\) has some dilation \(nX\) which is \(\varepsilon\)-dense in \(\mathbb T\).NEWLINENEWLINEHere generalizations to higher dimensions are given, where dilations are replaced by continuous endomorphisms \(A(n)\), \(\mathbb T^N\to\mathbb T^L\) with \(A(x)\in M_{L\times N}(\mathbb Z[x])\). Necessary and sufficient conditions are given for \(A\) such that any infinite subset \(X\) of \(\mathbb T^N\) is mapped by \(A(n)\) to an \(\varepsilon\)-dense subset of some translate of a subtorus of \(\mathbb T^L\). An estimate (depending on \(\varepsilon,N,L,A)\)) on the cardinality of \(X\) is given which guarantees the existence of such a \(n\in\mathbb N\).