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AbstractWe use bounds of character sums and some combinatorial arguments to show the abundance of very smooth numbers which also have very few nonzero binary digits.
Abstract Consider a minimal-free topological dynamical system $(X, \mathbb Z^d)$ . It is shown that the radius of comparison of the crossed product C*-algebra $\mathrm {C}(X) \rtimes \mathbb Z^d$ is at most half the mean topological dimension of $(X, \mathbb Z^d)$ . As a consequence, the C*-algebra $\mathrm {C}(X) \rtimes \mathbb Z^d$ is classified by the Elliott invariant if the mean dimension of $(X, \mathbb Z^d)$ is zero.