On finitely many base q expansions

2025/01/16 by Simon Baker, Baker, Simon, George Bender +1 · 1 citation
Computer Science · Mathematics · #Coding theory and cryptography #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical and Theoretical Analysis #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2501.09582

openalex publication_date 2025/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given some integer m ≥ 3, we find the first explicit collection of countably many intervals in (1,2) such that for any q in one of these intervals, the set of points with exactly m base q expansions is nonempty and moreover has positive Hausdorff dimension. Our method relies on an application of a theorem proved by Falconer and Yavicoli, which guarantees that the intersection of a family of compact subsets of ℝ^d has positive Hausdorff dimension under certain conditions.

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