Strong Characterizing Sequences in Simultaneous Diophantine Approximation

Biró, András and T. Sós, Vera (2003) Strong Characterizing Sequences in Simultaneous Diophantine Approximation. JOURNAL OF NUMBER THEORY, 99 (2). pp. 405-414. ISSN 0022-314X

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Abstract

Answering a question of Liardet. we prove that if 1, alpha(1), alpha(2)...... alpha(t) are real numbers linearly independent over the rationals, then there is an infinite subset A of the positive integers such that for real beta, we have (|| || denotes the distance to the nearest integer) Sigma(nequivalent toA)||nbeta||<infinity if and only if beta is a linear combination with integer coefficients of 1, alpha(1), alpha(2,)..., alpha(t). The proof combines elementary ideas with a deep theorem of Freiman on set addition. Using Freiman's theorem, we prove a lemma on the structure of Bohr sets, which may have independent interest. (C) 2002 Elsevier Science (USA). All rights reserved.

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