Hausdorff dimension of sets of numbers with large L\"uroth elements

Abstract

L\"uroth 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 L\"uroth series share several properties. In this paper, we explore one similarity by estimating the Hausdorff dimension of subsets of real numbers whose L\"uroth expansion grows at a definite rate. This is an extension of a result of Y. Sun and J. Wu to the context of L\"uroth 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.