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Mat. Zametki, 2005, Volume 77, Issue 6, Pages 803–813 (Mi mz2537)  

This article is cited in 5 scientific papers (total in 5 papers)

Density Modulo 1 of Sublacunary Sequences

R. K. Akhunzhanova, N. G. Moshchevitinb

a M. V. Lomonosov Moscow State University
b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We prove the existence of real numbers badly approximated by rational fractions whose denominators form a sublacunar sequence. For example, for the ascending sequence $s_n$, $n=1,2,3,…$, generated by the ordered numbers of the form $2^i3^j$, $i,j=1,2,3,…$, we prove that the set of real numbers $\alpha$, such that $\inf_{n\in\mathbb N}n\|s_n\alpha\|>0$, is a set of Hausdorff dimension 1. The divergence of the series $\sum_{n=1}^\infty\frac1n$ implies that the Lebesgue measure of those numbers is zero.

DOI: https://doi.org/10.4213/mzm2537

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English version:
Mathematical Notes, 2005, 77:6, 741–750

Bibliographic databases:

UDC: 511
Received: 17.02.2004

Citation: R. K. Akhunzhanov, N. G. Moshchevitin, “Density Modulo 1 of Sublacunary Sequences”, Mat. Zametki, 77:6 (2005), 803–813; Math. Notes, 77:6 (2005), 741–750

Citation in format AMSBIB
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  • https://doi.org/10.4213/mzm2537
  • http://mi.mathnet.ru/eng/mz/v77/i6/p803

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. N. G. Moshchevitin, “Sublacunary Sequences and Winning Sets”, Math. Notes, 78:4 (2005), 592–596  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. Dubickas A., “On the fractional parts of lacunary sequences”, Math. Scand., 99:1 (2006), 136–146  crossref  mathscinet  zmath  isi  elib
    3. Dubickas A., “An approximation by lacunary sequence of vectors”, Combin. Probab. Comput., 17:3 (2008), 339–345  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    4. N. G. Moshchevitin, “Khintchine's singular Diophantine systems and their applications”, Russian Math. Surveys, 65:3 (2010), 433–511  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. N. G. Moshchevitin, “Density modulo 1 of lacunary and sublacunary sequences: application of Peres–Schlag's construction”, J. Math. Sci., 180:5 (2012), 610–625  mathnet  crossref  mathscinet  elib
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