Maximally Spread Out Measures and Implications for Phase Transitions in Approximation Theory
Abstract
newcommandX𝕏newcommandSC𝒞 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⊂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∗:=liminfε↓ 0(loglog N(SC,ε))/log(1/ε) satisfies s∗>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∗ satisfies μ∗(X∖SC)= 0 and furthermore satisfies for every 0<s<s∗ the small-ball condition μ∗(B(x,r))≤exp(−c(s)·(1/r)s) for all x∈X and 0<r<r0(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∗ exists as an actual limit. There, critical measures were constructed for unit balls of certain Besov and Sobolev spaces considered as subsets of L2. In contrast, our construction is completely general. In particular, our results apply to function spaces of dominating mixed smoothness.
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BibTeX
@misc{matt2026maximally,
title = {{Maximally Spread Out Measures and Implications for Phase Transitions in Approximation Theory}},
author = {Hannes Matt and Erwin Riegler and Felix Voigtlaender},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30549},
eprint = {2608.30549},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30549},
}
RIS
TY - UNPB
TI - Maximally Spread Out Measures and Implications for Phase Transitions in Approximation Theory
AU - Hannes Matt
AU - Erwin Riegler
AU - Felix Voigtlaender
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30549
UR - https://arxiv.org/abs/2608.30549
AB - newcommandX𝕏newcommandSC𝒞 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⊂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_∗:=liminf_ε↓ 0(loglog N(SC,ε))/log(1/ε) satisfies s_∗>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^∗ satisfies μ^∗(X∖SC)= 0 and furthermore satisfies for every 0<s<s_∗ the small-ball condition μ^∗(B(x,r))≤exp(−c(s)·(1/r)^s) for all x∈X 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_∗ 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.
ER -