Publication Date:
2013-11-27
Description:
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 (that is, 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 x N matrix whose entries are in Z[ x ], and X be an infinite subset of T N . Contrary to the case N = L = 1, it is not always true that there is an integer n such that A ( n ) X is -dense in a translate of a subtorus of T L . We give a necessary and sufficient condition for matrices A for which this is true. We also prove an effective version of the result.
Print ISSN:
0024-6107
Electronic ISSN:
1469-7750
Topics:
Mathematics
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