2025/09/02 by Caleb Marshall, Marshall, Caleb
Mathematics · #11B75 (secondary) #11C08 #28A78 #28A80 (primary) #42B05 #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2509.02882
openalex publication_date 2025/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove a power law for the asymptotic decay of the Favard length of neighbourhoods of certain self-similar sets in ℝd with d ≥ 2. These self-similar sets are generalizations of the so-called four-corner Cantor set to higher dimensions, as well as to a more general class of rational digit sets. When d ≥ 3, our estimates are the first such non-trivial asymptotic upper bounds for the Favard length problem. The extension to a new class of digit sets (which is new even when d = 2, but holds for d ≥ 2 generally) uses the work of G. Kiss, I. Laba, G. Somlai and the author on vanishing sums of roots of unity and divisibility by many cyclotomic polynomials.