On the Intermediate Values of the Lower Quantization Dimension

IF 0.6 4区数学 Q3 MATHEMATICS

Mathematical Notes Pub Date : 2024-07-05 DOI:10.1134/s0001434624030039

A. V. Ivanov

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Abstract

It is well known that the lower quantization dimension \(\underline{D}(\mu)\) of a Borel probability measure \(\mu\) given on a metric compact set \((X,\rho)\) does not exceed the lower box dimension \(\underline{\dim}_BX\) of \(X\). We prove the following intermediate value theorem for the lower quantization dimension of probability measures: for any nonnegative number \(a\) smaller that the dimension \(z\underline{\dim}_BX\) of the compact set \(X\), there exists a probability measure \(\mu_a\) on \(X\) with support \(X\) such that \(\underline{D}(\mu_a)=a\). The number \(z\underline{\dim}_BX\) characterizes the asymptotic behavior of the lower box dimension of closed \(\varepsilon\)-neighborhoods of zero-dimensional, in the sense of \(\dim_B\), closed subsets of \(X\) as \(\varepsilon\to 0\). For a wide class of metric compact sets, the equality \(z\underline{\dim}_BX=\underline{\dim}_BX\) holds.

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关于低量化维度的中间值

Abstract 众所周知，给定在度量紧凑集\((X,\rho)\)上的博尔概率度量的下量子化维度（(\underline{D}(\mu)\）不超过\(X\)的下盒维度（(\underline{dim}_BX\）。我们证明了以下关于概率度量的下量化维度的中间值定理：对于任何小于紧凑集\(X\)的维度\(z/underline{/dim}_BX\) 的非负数\(a\)，在\(X\)上存在一个概率度量\(\mu_a\)，其支持度为\(X\)，使得\(\underline{D}(\mu_a)=a\)。数字\(z\underline{dim}_BX\) 描述了封闭的\(\varepsilon\)-零维邻域的下盒维的渐近行为，在\(\dim_B\)的意义上，\(X\)的封闭子集为\(\varepsilon\to 0\).对于很宽的一类度量紧凑集来说，等式（z（underline{/dim}_BX=underline{/dim}_BX）成立。

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来源期刊

Mathematical Notes 数学-数学

CiteScore

0.90

自引率

16.70%

发文量

179

审稿时长

24 months

期刊介绍： Mathematical Notes is a journal that publishes research papers and review articles in modern algebra, geometry and number theory, functional analysis, logic, set and measure theory, topology, probability and stochastics, differential and noncommutative geometry, operator and group theory, asymptotic and approximation methods, mathematical finance, linear and nonlinear equations, ergodic and spectral theory, operator algebras, and other related theoretical fields. It also presents rigorous results in mathematical physics.