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

$$ We establish the existence of a “maximally spread out” Borel probability measure on a totally bounded subset $\text{SC}$ of a (quasi)-Banach space $\text{X}$ under two mild conditions: (i) a growth condition on the covering numbers $N(\text{SC},\epsilon )$ of $\text{SC}$, and (ii) a technical topological condition that is in particular satisfied whenever $\text{SC}\subset \text{X}$ is closed, bounded, and convex. More formally, condition (i) requires that the so-called lower power-exponential Minkowski dimension of $\text{SC}$, i.e., $$s_{\ast }:=\underset{\epsilon \downarrow 0}{\mathrm{lim\hspace{0.17em}inf}}\frac{\log \log N(\text{SC},\epsilon )}{\log (1\mathrm{/}\epsilon )}$$ satisfies $s_{\ast }>0$. Under these conditions, we construct a Borel probability measure $\mu$ on $\text{X}$ that is critical for $\text{SC}$, or maximally spread out, meaning that the associated outer measure ${\mu }^{\ast }$ satisfies ${\mu }^{\ast }(\text{X}\setminus \text{SC})=0$ and furthermore satisfies for every $0<s<s_{\ast }$ the small-ball condition $${\mu }^{\ast }(B(x,r))\le \exp (-c(s)\cdot (1\mathrm{/}r{)}^s)\text{ for all }x\in \text{X}\text{ and }0<r<r_0(s)\mathrm{.}$$ The existence of such a critical measure in particular implies that the so-called power-exponential Hausdorff dimension of $\text{SC}$ 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 $\text{SC}$, provided that the $\mathrm{lim\hspace{0.17em}inf}$ 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.