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An approximation by lacunary sequence of vectors. (English) Zbl 1221.11164

Summary: Let \((t_k)_{k=0}^{\infty}\) be a sequence of real numbers satisfying \(t_0 \neq 0\) and \(|t_{k+1}| \geq (1+1/M) |t_{k}|\) for each \(k\geq 0\), where \(M\geq 1\) is a fixed number. We prove that, for any sequence of real numbers \((\xi_k)_{k=0}^{\infty}\), there is a real number \(\xi\) such that \(\|t_k \xi-\xi_k\|>1/(80M \log(28M))\) for each \(k\geq 0\). Here, \(\|x\|\) denotes the distance from \(x \in \mathbb R\) to the nearest integer. This is a corollary derived from our main theorem, which is a more general matrix version of this statement with explicit constants.

MSC:

11K31 Special sequences
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[1] Flatto, Acta Arith. 70 pp 125– (1995)
[2] Erd?s, Repartition Modulo 1, Actes Colloq. Marseille?Luminy 1974 pp 89– (1975)
[3] Dubickas, Math. Scand. 99 pp 136– (2006) · Zbl 1140.11038 · doi:10.7146/math.scand.a-15004
[4] DOI: 10.1112/S0024609305017728 · Zbl 1164.11025 · doi:10.1112/S0024609305017728
[5] DOI: 10.1007/BF01475864 · JFM 46.0278.06 · doi:10.1007/BF01475864
[6] DOI: 10.1007/s004930100019 · Zbl 0981.05038 · doi:10.1007/s004930100019
[7] Pollington, Illinois J. Math. 23 pp 511– (1979)
[8] DOI: 10.1007/s11006-005-0075-2 · Zbl 1078.11048 · doi:10.1007/s11006-005-0075-2
[9] Akhunzhanov, Dokl. Ross. Akad. Nauk 397 pp 295– (2004)
[10] DOI: 10.1007/BF01898138 · Zbl 0465.10040 · doi:10.1007/BF01898138
[11] Khintchine, Rend. Circ. Mat. Palermo 50 pp 170– (1926)
[12] Alon, The Probabilistic Method (1992)
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