Decomposition of real numbers into sums of Lüroth sets
Maiken Gravgaard, Ying Wai Lee
TL;DR
The paper investigates decompositions of real numbers as sums of Lüroth sets defined by digit constraints, extending classical continued-fraction results to Lüroth expansions. It develops a framework of Stepwise Complete Constructions to realize Lüroth sets as general Cantor sets, computes thickness and gamma, and applies Cantor-sum theory (Hall–Hlavka–Astels–Peres–Shmerkin) to obtain both congruence modulo 1 results and Hausdorff-dimension estimates for Lüroth sumsets. Key contributions include explicit congruence results such as $L_{ rianglelefteq 3}+L_{ rianglelefteq 5}\models ext{R mod }1$, as well as dimension formulas and bounds: for finite digit-sets $ riangle A, riangle B$, $ ext{dim}(L_{ riangle A}+L_{ riangle B})= ext{min}(1, ext{dim}(L_{ riangle A})+ ext{dim}(L_{ riangle B}))$, and $ ext{dim}(L_{ riangleright k})> frac{1}{2}+ frac{1}{2 ext{log max}\{16,k ight\}}$, with $ ext{dim}(L_{ riangleright k}+ obreak L_{ riangleright k})=1$. These results mirror and extend the continued-fraction literature to Lüroth expansions, highlighting rich additive and fractal structure with potential applications in Diophantine approximation and dynamical systems.
Abstract
We study the decomposition of real numbers into sums of Lüroth sets, which are defined by numbers whose Lüroth expansions have prescribed digit constraints. We establish several results on the congruence modulo 1 of sums of Lüroth sets, including summands with digits bounded above, below, and combinations of the two. We also analyse the Hausdorff dimension of Lüroth sets and their sums. The results extend classical findings on continued fractions to Lüroth expansions.