S(x) satisfies S(x)/(x (log x)^R) → +∞ as x → ∞ for every fixed R > 0, resolving Erdős Problem 1061.
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Hub-and-spoke systems from symbolic dynamics can have completely positive mean dimension without uniformly positive mean dimension or entropy, with proofs linking entropy and mean dimension properties at the level of fixed covers.
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Uniformly Positive Mean Dimension
Hub-and-spoke systems from symbolic dynamics can have completely positive mean dimension without uniformly positive mean dimension or entropy, with proofs linking entropy and mean dimension properties at the level of fixed covers.