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Strong characterizing sequences in simultaneous Diophantine approximation. (English) Zbl 1058.11047

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 [A. Biró, J.-M. Deshouillers, and 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).

MSC:

11J13 Simultaneous homogeneous approximation, linear forms
11P70 Inverse problems of additive number theory, including sumsets

Citations:

Zbl 1006.11038
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References:

[1] Biró, A.; Deshouillers, J.-M.; Sós, V. T., Good approximation and characterization of subgroups of \(R/Z\), Studia Sci. Math. Hungar., 38, 97-113 (2001) · Zbl 1006.11038
[2] P. Liardet, oral communication.; P. Liardet, oral communication.
[3] Kraaikamp, C.; Liardet, P., Good approximations and continued fractions, Proc. Amer. Math. Soc., 112, 2, 303-309 (1991) · Zbl 0725.11033
[4] Petersen, K., On a series of cosecants related to a problem in ergodic theory, Compositio Math., 26, 3, 313-317 (1973) · Zbl 0269.10030
[5] Ruzsa, I. Z., Generalized arithmetical progressions and sumsets, Acta Math. Hungar., 65, 4, 379-388 (1994) · Zbl 0816.11008
[6] Freiman, G., Foundations of a Structural Theory of Set Addition, Translations of Mathematical Monographs, Vol. 37 (1973), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0271.10044
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