Maximally Spread Out Measures and Implications for Phase Transitions in Approximation Theory

Abstract

$\newcommand{\X}{\mathbb{X}}\newcommand{\SC}{\mathcal{C}}$ We establish the existence of a "maximally spread out" Borel probability measure on a totally bounded subset $\SC$ of a (quasi)-Banach space $\X$ under two mild conditions: (i) a growth condition on the covering numbers $N(\SC, ε)$ of $\SC$, and (ii) a technical topological condition that is in particular satisfied whenever $\SC\subset\X$ is closed, bounded, and convex. More formally, condition (i) requires that the so-called lower power-exponential Minkowski dimension of $\SC$, i.e., \[s_\ast:=\liminf_{ε\downarrow 0}\frac{\log\log N(\SC,ε)}{\log(1/ε)}\] satisfies $s_\ast>0$. Under these conditions, we construct a Borel probability measure $μ$ on $\X$ that is critical for $\SC$, or maximally spread out, meaning that the associated outer measure $μ^\ast$ satisfies $μ^\ast(\X\setminus\SC)= 0$ and furthermore satisfies for every $0<s<s_\ast$ the small-ball condition \[μ^\ast(B(x,r))\le\exp\bigl(-c(s)\cdot(1/r)^s\bigr)\quad\text{ for all }x\in\X\text{ and }0<r<r_0(s).\] The existence of such a critical measure in particular implies that the so-called power-exponential Hausdorff dimension of $\SC$ introduced in [J.~Topol.~Anal.~4(2):203--235, 2012] coincides with the lower power-exponential Minkowski dimension. Previous work [Found.~Comput.~Math.~23(1):329--392, 2023] shows that such a critical measure gives rise to a phase transition regarding lossy compression and approximation by quantized neural networks of elements of $\SC$, provided that the $\liminf$ in the definition of $s_\ast$ exists as an actual limit. There, critical measures were constructed for unit balls of certain Besov and Sobolev spaces considered as subsets of $L^2$. In contrast, our construction is completely general. In particular, our results apply to function spaces of dominating mixed smoothness.