Scales
Mathieu Helfter
TL;DR
The paper develops a unified framework of scales that generalizes classical dimension theory to infinite-dimensional metric spaces and measures by introducing a scaling family $\mathsf{scl}= (\mathrm{scl}_\alpha)_{\alpha>0}$. It defines metric and measure scales (Hausdorff, packing, box, local, and quantization) under these scalings and proves general comparison theorems linking the different scales; in particular, it extends known dimension inequalities to the scale setting. The authors apply the theory to Wiener measure, C^0-functional spaces, and ergodic decomposition emergence, obtaining sharp equalities and inequalities among various orders and solving Berger’s emergence problem in the local-to-quantization context. They also provide detailed constructions and examples, including infinite products of finite sets, to illustrate how scales behave in complex, infinite-dimensional spaces. The framework offers a robust toolkit for analyzing the geometric and probabilistic structure of infinite-dimensional objects, with potential implications for dynamical systems, functional analysis, and probability.
Abstract
We introduce the notion of scale to generalize and compare different invariants of metric spaces and their measures. Several versions of scales are introduced such as Hausdorff, packing, box, local and quantization. They moreover are defined for different growth, allowing in particular a refined study of infinite dimensional spaces. We prove general theorems comparing the different versions of scales. They are applied to describe geometries of ergodic decompositions, of the Wiener measure and of functional spaces. The first application solves a problem of Berger on the notions of emergence (2020); the second lies in the geometry of the Wiener measure and extends the work of Dereich-Lifshits (2005); the last refines Kolmogorov-Tikhomirov (1958) study on functional spaces.