Decomposition of real numbers into sums of L\"uroth sets

TL;DR

This paper investigates how real numbers can be expressed as sums of L"uroth sets with specific digit constraints, analyzing their properties, dimensions, and extending classical results from continued fractions to L"uroth expansions.

Contribution

It introduces new results on the decomposition of real numbers into sums of L"uroth sets and explores their Hausdorff dimensions, extending classical continued fraction theory.

Findings

Results on congruence modulo 1 of sums of L"uroth sets

Hausdorff dimension analysis of L"uroth sets and their sums

Extension of classical continued fraction results to L"uroth expansions

Abstract

We study the decomposition of real numbers into sums of L\"uroth sets, which are defined by numbers whose L\"uroth expansions have prescribed digit constraints. We establish several results on the congruence modulo 1 of sums of L\"uroth sets, including summands with digits bounded above, below, and combinations of the two. We also analyse the Hausdorff dimension of L\"uroth sets and their sums. The results extend classical findings on continued fractions to L\"uroth expansions.