Strong characterizing sequences in simultaneous diophantine approximation

AbstractAnswering a question of Liardet, we prove that if 1,α1,α2,…,αt are real numbers linearly independent over the rationals, then there is an infinite subset A of the positive integers such that for real β, we have (|||| denotes the distance to the nearest integer)∑n∈A||nβ||<∞if and only if β is a linear combination with integer coefficients of 1,α1,α2,…,αt. 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