Continued fractions and heavy sequences

Abstract

We initiate the study of the sets H(c), 0 c). More formally, H(c) def {α∈ℝ | card ({1≤k≤ n | (kα) cn, for all n > 1}, where (x) = x ― [x] stands for the fractional part of x ∈ R. We prove that, for rational c, the sets H(c) are of positive Hausdorff dimension and, in particular, are uncountable. For integers m > 1, we obtain a surprising characterization of the numbers α ∈ H m = def H(1/m) in terms of their continued fraction expansions: The odd entries (partial quotients) of these expansions are divisible by m. The characterization implies that x ∈ H m if and only if 1/mx ∈ H m , for x > 0. We are unaware of a direct proof of this equivalence without making use of the mentioned characterization of the sets H m . We also introduce the dual sets H m of reals y for which the sequence of integers ([ky]) k≥1 consistently hits the set mZ with the at least expected frequency 1/m and establish the connection with the sets H m : If xy = m for x, y > 0, then x ∈ Hm ⇒ y ∈ H m . The motivation for the present study comes from Y. Peres's ergodic lemma.