Mean dimension of function classes with Lebesgue measurable spectral sets (Q1308789)

The notion of mean dimension which was introduced by V. M. Tikhomirov in the 1970's is close to that of the mean entropy. Both of them constitute the mean amount of information which is necessary to identify an element of the given function class. The mean dimension shows how economically we can approximate function classes by finite-dimensional linear subspaces. Tikhomirov [see ``Encyclopedic of Mathematical Sciences'', Vol. 14, Analysis II. Convex Analysis and Approximations (1990)] posed the following problem: find the mean dimension of the unit ball \(B^ p_ E\) of the space of \(L^ p\)-functions on \(\mathbb{R}^ n\) with spectra inside a given Lebesgue measurable bounded subset \(E\) of \(\mathbb{R}^ n\). In this paper, this problem posed by Tikhomirov, known as Tikhomirov conjecture on mean dimension, is confirmed in certain cases and shown to fail in certain other cases.