最大扩散测度及其对逼近论中相变的影响
Maximally Spread Out Measures and Implications for Phase Transitions in Approximation Theory
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中文总结 AI 辅助
该研究在温和条件下证明了拟巴拿赫空间完全有界子集上最大扩散博雷尔概率测度的存在性，统一了幂指数豪斯多夫维数与下幂指数闵可夫斯基维数，其通用构造可应用于支配混合光滑度函数空间的逼近论相变研究。
中文摘要 AI 辅助
我们在两个温和条件下，证明了（拟）巴拿赫空间$\newcommand{\u005cX}{\u005cmathbb{X}}\newcommand{\u005cSC}{\u005cmathcal{C}}\u005cX$的完全有界子集$\u005cSC$上“最大扩散”博雷尔概率测度的存在性：（i）关于$\u005cSC$的覆盖数$N(\u005cSC, \u005cvarepsilon)$的增长条件；（ii）一个技术性拓扑条件，当$\u005cSC\u005csubset\u005cX$是闭、有界且凸集时，该条件必然满足。\n更正式地说，条件（i）要求$\u005cSC$的所谓下幂指数闵可夫斯基维数，即\n$$s_\u005cast:=\u005climinf_{\u005cvarepsilon\u005cdownarrow 0}\u005cfrac{\u005clog\u005clog N(\u005cSC,\u005cvarepsilon)}{\u005clog(1/\u005cvarepsilon)}$$\n满足$s_\u005cast>0$。在这些条件下，我们构造了$\u005cX$上的一个博雷尔概率测度$\u005cmu$，它对$\u005cSC$而言是临界的，或者说是最大扩散的，即相关的外测度$\u005cmu^\u005cast$满足$\u005cmu^\u005cast(\u005cX\u005csetminus\u005cSC)= 0$，并且对于每个$0<s<s_\u005cast$，满足小球条件：\n$$\u005cmu^\u005cast(B(x,r))\u005cle\u005cexp\u005cbigl(-c(s)\u005ccdot(1/r)^s\u005cbigr)\u005cquad\u005ctext{ 对所有 }x\u005cin\u005cX\u005ctext{ 和 }0<r<r_0(s)\u005ctext{ 成立}。$$\n这种临界测度的存在性尤其意味着，发表于《J. Topol. Anal.》2012年第4卷第2期203-235页的论文中引入的$\u005cSC$的幂指数豪斯多夫维数，与下幂指数闵可夫斯基维数一致。\n发表于《Found. Comput. Math.》2023年第23卷第1期329-392页的前期研究表明，若$s_\u005cast$定义中的下极限是真实存在的极限，则这种临界测度会引发关于$\u005cSC$中元素的有损压缩以及量化神经网络逼近的相变。该研究针对被视为$L^2$子集的某些Besov空间和Sobolev空间的单位球构造了临界测度。相比之下，我们的构造是完全通用的，尤其适用于支配混合光滑度的函数空间。
英文摘要
$\newcommand{\X}{\mathbb{X}}\newcommand{\SC}{\mathcal{C}}$ 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.
