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 2i3j, i, j = 1, 2, 3, ..., we prove that the set of real numbers α such that inf n∈ℕ n‖s n α‖ > 0 is a set of Hausdorff dimension 1. The divergence of the series \(\sum _{n = 1}^\infty \tfrac{1}{n}\) implies that the Lebesgue measure of those numbers is zero.
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Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 803–813.
Original Russian Text Copyright ©2005 by R. K. Akhunzhanov, N. G. Moshchevitin.
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Akhunzhanov, R.K., Moshchevitin, N.G. Density Modulo 1 of Sublacunary Sequences. Math Notes 77, 741–750 (2005). https://doi.org/10.1007/s11006-005-0075-2
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DOI: https://doi.org/10.1007/s11006-005-0075-2