2024/11/25 by Jing Yu, Xingyu Zhu, Yu, Jing +1
Computer Science · #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2411.16660
openalex publication_date 2024/11/25 · openalex created_date 2024/12/04 · openalex updated_date 2026/07/28
We develop a probabilistic framework for large-scale dimension bounds in metric geometry, based on padded decompositions, randomized ball carving on net graphs, and the Lovász Local Lemma. For metric measure spaces with volume doubling constant C\mathsf D, we prove the sharp bound asdimAN(X)≤ dimAN(X)≤ \lfloorlog2 C\mathsf D\rfloor. In particular, if (M,g) is a complete Riemannian n-manifold with Ricg≥ 0, then asdim(M)≤ n, thereby settling a question of Papasoglu on manifolds with nonnegative Ricci curvature. We also show that if (X,d,\mathfrakm) is proper, volume noncollapsed, and has polynomial volume growth rate ρV(X), then asdim(X)≤ \lfloorρV(X)\rfloor. Moreover, the corresponding control function can be chosen to have polynomial growth. This extends Papasoglu's sharp asymptotic-dimension bound from graphs of polynomial growth to a metric-measure setting. As applications, we study equality in the polynomial-growth bound for universal covers of nilmanifolds, and under nonnegative Ricci curvature we relate the equality case in the volume-doubling bound to Gromov largeness, obtaining in particular a consequence for complete manifolds with positive scalar curvature.