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
implies that the Lebesgue measure of those numbers is zero.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 803–813.Original Russian Text Copyright ©2005 by R. K. Akhunzhanov, N. G. Moshchevitin. |