Decomposition of real numbers into sums of Lüroth sets

2025/06/14 by Gravgaard, Maiken, Lee, Ying Wai · 1 citation
#11J70 (Primary) #11J83 #11K55 #28A80 #37A05 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2506.12513

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.

Citations

• Every real number is a sum of two real numbers with diverging partial quotients
• Explicit Upper Bounds on Decay Rates of Fourier Transforms of Self-similar Measures on Self-similar Sets
• On a Conjecture of Cusick on a sum of Cantor sets
• Fourier transform and expanding maps on Cantor sets
• Metric Diophantine Approximation: aspects of recent work
• Resonance between Cantor sets
• Additive functions of intervals and Hausdorff measure

Cited by

• Representations of rational numbers and Minkowski dimension

Related

• Hausdorff dimension of sets of numbers with large Lüroth elements
• Hausdorff dimension of some exceptional sets in Lüroth expansions
• Bounded Lüroth expansions: applying Schmidt games where infinite distortion exists
• A note on Diophantine-type fractals for α-Lüroth systems
• On the Hausdorff dimension of certain sets arising in Number Theory