Strong characterizing sequences in simultaneous Diophantine approximation.
In this paper, it is proved that: if \(1,\alpha_1,\dots, \alpha_t\in\mathbb{R}\) are linearly independent over the rationals, there is a subset \(A\subset\mathbb{N}\), \(| A|=\infty\), such that \(\sum_{n\in A}\| n\beta\|\) is finite if and only if \(\beta\in G\), the group generated by \(1,\alpha_1,\dots, \alpha_t\). Moreover, if \(\beta\not\in G\), then we even have \(\liminf_{n\in A,n\to+\infty}\| n\beta\|> 0\). This strengthens a theorem proved in [\textit{A. Biró}, \textit{J.-M. Deshouillers}, and \textit{V. T. Sós}, Stud. Sci. Math. Hung. 38, 97--113 (2001; Zbl 1006.11038)]. The proof uses both ideas from this paper and Freiman's theory (in the form of a structural lemma on Bohr sets).
scientific article; zbMATH DE number 1915514