Uniform dilations in higher dimensions

نویسندگان

  • Michael Kelly
  • Thái Hoàng Lê
چکیده

A theorem of Glasner says that if X is an infinite subset of the torus T, then for any > 0, there exists an integer n such that the dilation nX = {nx : x ∈ T} is -dense (i.e, it intersects any interval of length 2 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 T . Contrarily to the case N = L = 1, it’s not always true that there is an integer n such that A(n)X is -dense in a translate of a subtorus of T. We give a necessary and sufficient condition for matrices A for which this is true. We also prove an effective version of the result.

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عنوان ژورنال:
  • J. London Math. Society

دوره 88  شماره 

صفحات  -

تاریخ انتشار 2013