Beatty sequences and multiplicative number theory
On the Hausdorff dimension of certain self-affine sets
A subset E of ℝⁿ is called self-affine with respect to a collection ϕ₁,...,ϕₜ of affinities if E is the union of the sets ϕ₁(E),...,ϕₜ(E). For S ⊂ ℝⁿ let . If Φ(S) ⊂ S let denote . For given Φ consisting of contracting “pseudo-dilations” (affinities which preserve the directions of the coordinate axes) and subject to further mild technical restrictions we show that there exist self-affine sets of each Hausdorff dimension between zero and a positive number depending on Φ. We also investigate...
Arithmetic functions on Beatty sequences
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