Unexpected distribution phenomenon resulting from Cantor series expansions

Abstract

We explore in depth the number theoretic and statistical properties of certain sets of numbers arising from their Cantor series expansions. As a direct consequence of our main theorem we deduce numerous new results as well as strengthen known ones.

An analogue of Pillai's theorem for continued fraction normality and an application to subsequences

“…This result contrasts with the behavior of base-b normality, where such a subsequence is always normal [17]. Settings under which normality is conditionally preserved along arithmetic progressions were shown by Mance and Airey in [1,2]. Heersink and Vandehey employ an ergodic-theoretic argument, whereas our proof is elementary and combinatorial.…”

“…)). By Lemma 5.2, this is not equal to γ(C [1,1] ) and from Theorem 4.1 it follows that the subsequence is nonnormal.…”

“…Thus the asymptotic frequency of the pattern [1,1] in x is not γ(C [1,1] ), proving that x is not continued-fraction normal.…”

“…Note that T i x ∈ C w if and only if w appears at the (i + 1)th position in the continued fraction expansion of x. In this notion, we consider overlapping occurrences of w as distinct -for example, if w is [1,1], then it occurs twice in the subsequence [1,1,1]. Since T is ergodic with respect to the Gauss measure, by the ergodic theorem, it follows that the set of continued fraction normals has Gauss measure 1.…”

Shrinking Targets for Nonautonomous Dynamical Systems Corresponding to Cantor Series Expansions

“…Later, certain aspects of this area were extensively studied by many authors, notably by Galambos [10], Salát (see for example [19]) and Schweiger [20]. Most recently, the second author and his collaborators continued to develop this area of research in [1,2,14], where the primary results concern various concepts of normality, relations between them and the Hausdorff dimensions of appropriate significant sets.…”