On the intermediate values of the box dimensions (Q6101505)

The box dimension \(\dim_B X\) of a metric compactum \((X,\rho)\) is defined by the formula \[ \dim_B X=\lim_{\varepsilon\to 0}\frac{\log N(X,\varepsilon)}{-\log\varepsilon}, \] where \(N(X,\varepsilon)\) denotes the least number of closed balls of radius \(\varepsilon\) needed to cover \(X\). If the limit does not exist, then we use the upper and lower limits, yielding the upper box dimension \(\overline{\dim}_B X\) and the lower box dimension \(\underline{\dim}_B X\) of \(X\). It was established in [\textit{Y. B. Pesin}, Dimension theory in dynamical systems: contemporary views and applications. Chicago: The University of Chicago Press (1998; Zbl 0895.58033)] that on the segment \(I=[0, 1]\subset\mathbb{R}\), for every two reals \(c\) and \(b\) such that \(0\leqslant c\leqslant b\leqslant 1\), there exists a closed subset \(A\subset I\) for which \(\underline{\dim}_B A=c\) and \(\overline{\dim}_B A=b\) (but \(\dim_B I=1\)). The above result motivates the general question: \textbf{Question.} Let \(X\) be a metric compactum and \(\dim_B X=a\leqslant\infty\). Is it true that for all reals \(c\) and \(b\) such that \(0\leqslant c\leqslant b\leqslant a\), there is a closed subset \(A\subset X\) satisfying \(\underline{\dim}_B A=c\) and \(\overline{\dim}_B A=b\)? It was shown in [\textit{A. V. Ivanov} and \textit{O. V. Fomkina}, ``On the order of metric approximation of maximal linked systems and capacitarian dimensions'', Trans. KarRC RAS 7, 5--14 (2019; \url{doi: 10.17076/mat1034})] that if \(\overline{\dim}_B X=a\leqslant\infty\), then for every \(b\in[0,a]\) there exists a closed subset \(A\subset X\) such that \(\overline{\dim}_B A=b\). In this paper, the author proves the following intermediate value theorem for the box dimension: \textbf{Theorem.} Let \((X,\rho)\) be a metric compactum with \(\overline{\dim}_B X=a\leqslant\infty\). Then for every nonnegative real \(b\leqslant a\) there is a closed subset \(A\subset X\) satisfying \(\overline{\dim}_B A=b\) and \(\underline{\dim}_B A=0\). Thus, the author answers the above question in the positive for \(c=0\). In the general case, this result is final. The author constructs an example of a metric compactum \(Z\) (homeomorphic to a Cantor perfect set) such that \(\dim_B Z=1\) and \(\underline{\dim}_B F=0\) for every nonempty proper closed subset \(F\subset Z\).