On the quantization dimension of maximal linked systems (Q6610181)

A covariant functor \(\mathcal{F}\) in the category \(Comp\) of compact spaces and their continuous mappings is seminormal if \(\mathcal{F}\) is monomorphic and continuous as well as preserving intersections, a point, and the empty set. For a seminormal functor \(\mathcal{F}\), a compact space \(X\), and \(\xi\in \mathcal{F}(X)\), the notion of support \(\operatorname{supp}(X)\) is defined as the inclusion least \(A\subset X\) such that \(\xi\in \mathcal{F}(A)\), put \N\[\N\mathcal{F}_{n}(X)=\left\{\xi\in \mathcal{F}(X): \operatorname{supp} (\xi)\leq n \right\}.\N\]\NLet \(\mathcal{F}\) be a seminormal weight-preserving metrizable functor in the category \(Comp\) of compact spaces and their continuous mappings and let, \((X, \rho)\) be a compact metric space. Then the upper and lower quantization dimensions \(\overline{\dim}_{\mathcal{F}}\xi\) and \(\underline{\dim}_{\mathcal{F}}\xi\) are defined for every point \(\xi\in \mathcal{F}(X)\).\N\NRecall that \(F\subset X\) is \(\varepsilon\)-separated for some \(\varepsilon>0\) if \(\rho(x, y)>\varepsilon\) for every two different points \(x, y\in F.\) Every \(\varepsilon\)-separated set in a compact space is finite. A subset \(F\subset X\) is an \(\varepsilon\)-net for \(X\) if \(B(F, \varepsilon)= X.\)\N\NIn this paper the following results are proved:\N\N\textbf{Lemma 1.} If \(A=\left\{x_{n}: n\in \mathbb{N}\right\}\) and \(B=\left\{y_{n}: n\in \mathbb{N}\right\}\) are two disjoint sequences of points in \(X\) such that \(\overline{A}\cap \overline{B}\neq\emptyset\) and for some \(k\in \mathbb{N}\) and \(\varepsilon>0\), the set \(D=\left\{x_{1},\dots, x_{k+1}\right\}\cup \left\{y_{1},\dots, y_{k}\right\}\) is \(\varepsilon\)-separated then \(N(\xi(A, B), \varepsilon/2)\geq 2k\).\N\N\textbf{Lemma 2.} If \(A=\left\{x_{n}: n\in \mathbb{N}\right\}\) and \(B=\left\{y_{n}: n\in \mathbb{N}\right\}\) are two disjoint sequences of points in \(X\) such that \(\overline{A}\cap \overline{B}\neq\emptyset\) and the inequality \(\rho(x_{k+1}, y_{k+1})\leq \varepsilon\) holds for some \(k\in \mathbb{N}\) and \(\varepsilon>0\); then \(N(\xi(A, B), \varepsilon)\geq 2k+1\).\N\N\textbf{Proposition 2.} If \((X, \rho)\) is a compact metric space, then there is an \(mls \xi\) satisfying\N\[\N\underline{\dim}_{\lambda}\xi=\underline{\dim}_{B}X, \overline{\dim}_{\lambda}\xi=\overline{\dim}_{B}X, \operatorname{supp}(\xi)=X.\N\]\N\textbf{Theorem 3.} Let \((X, \rho)\) be a compact metric space. For every nonnegative real \(b\leq\underline{\dim}_{B}X=a\leq\infty,\) there exists an \(mls \xi\in \lambda X\) for which \(\underline{\dim}_{\lambda}(\xi)=b\) and \(\operatorname{supp}(\xi)=X\).