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

TL;DR

This paper estimates the Hausdorff dimension of sets of real numbers with L"uroth expansions growing at a specific rate, extending previous results from continued fractions to L"uroth series.

Contribution

It extends the Hausdorff dimension estimation from continued fractions to L"uroth series, providing new insights into their measure-theoretic properties.

Findings

Hausdorff dimension bounds for L"uroth expansion sets

Extension of Sun and Wu's results to L"uroth series

Recent proof that the lower bound is an exact value

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.