Uniform Dilations in Higher Dimensions

Original · EN

A theorem of Glasner says that if X is an infinite subset of the torus T, then for any ε>0, there exists an integer n such that the dilation nX={nx: x ∈ T } is ε-dense (i.e, it intersects any interval of length 2ε in T). Alon and Peres provided a general framework for this problem, and showed quantitatively that one can restrict the dilation to be of the form f(n)X where f ∈ Z[x] is not constant. Building upon the work of Alon and Peres, we study this phenomenon in higher dimensions. Let A(x) be an L × N matrix whose entries are in Z[x], and X be an infinite subset of Tⁿ. Contrarily to the case N=L=1, it's not always true that there is an integer n such that (n)X is ε-dense in a translate of a subtorus of Tˡ. We give a necessary and sufficient condition for matrices A for which this is true. We also prove an effective version of the result.