2024/12/29 by Richárd Balka, Balka, Richárd, Vilma Orgoványi +3
Computer Science · #28A80 (Primary) 28A78 (Secondary) #Bayesian Methods and Mixture Models #Dynamical Systems (math.DS) #FOS: Mathematics #Metaheuristic Optimization Algorithms Research #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2412.20594
openalex publication_date 2024/12/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We resolve a few questions regarding the uniformity and size of microsets of subsets of Euclidean space. First, we construct a compact set K⊂ℝd with Assouad dimension arbitrarily close to d such that every microset of K has no Ahlfors--David regular subset with dimension strictly larger than 0. This answers a question of Orponen. Then, we show that for any non-empty compact set K⊂ℝd with lower dimension β, there is a microset E of K with finite β-dimensional packing pre-measure. This answers a strong version of a question of Fraser--Howroyd--Käenmäki--Yu, who previously obtained a similar result concerning the upper box dimension.