Hausdorff dimension of sets of numbers with large Lüroth elements
arXiv:2010.13932
Abstract
Lüroth series, like regular continued fractions, provide an interesting identification of real numbers with infinite sequences of integers. These sequences give deep arithmetic and measure-theoretic properties of subsets of numbers according to their growth. Although different, regular continued fractions and Lüroth series share several properties. In this paper, we explore one similarity by estimating the Hausdorff dimension of subsets of real numbers whose Lüroth expansion grows at a definite rate. This is an extension of a result of Y. Sun and J. Wu to the context of Lüroth series. It was recently shown by Y. Feng, B. Tan, and Q.-L. Zhou that the lower bound in our main theorem is actually an equality.
arXiv comment18 pages, 0 figures. We added the proof of Lemma 5.4, fixed several typos, improved the introduction, and added the reference to a recent paper by Feng, Tan, and Zhou. There are no changes in the main results. Accepted by Integers