Cantor sets for regular continued fractions and Lüroth series

Abstract

In this thesis, we consider the summation of Cantor sets. After a brief introduction to these sets, specifically focusing on the Cantor Middle Third set, we explore the relevance of Cantor sets in various number expansions, including $r$-ary expansions, regular continued fraction expansions, and L\"uroth series expansions. Marshall Hall, an American mathematician, extensively studied regular continued fractions, leading to significant discoveries, such as his theorem that every real number can be expressed as the sum of two regular continued fractions with partial quotients less than or equal to $4$. This thesis extends Hall's investigations to L\"uroth series expansions, aiming to establish analogous results to Hall's theorem.