Projections of four corner Cantor set: Total self-similarity, spectrum and unique codings

Abstract

Given ρ∈(0,1/4], the four corner Cantor set E⊂R2 is a self-similar set generated by the iterated function system {(ρx,ρy),(ρx,ρy+1−ρ),(ρx+1−ρ,ρy),(ρx+1−ρ,ρy+1−ρ)}. For θ∈[0,π) let Eθ be the orthogonal projection of E onto a line with an angle θ to the x-axis. In principle, Eθ is a self-similar set having overlaps. In this paper we give a complete characterization on which the projection Eθ is totally self-similar. We also study the spectrum of Eθ, which turns out that the spectrum achieves its maximum value if and only if Eθ is totally self-similar. Furthermore, when Eθ is totally self-similar, we calculate its Hausdorff dimension and study the subset Uθ which consists of all x∈Eθ having a unique coding. In particular, we show that dimHUθ=dimHEθ for Lebesgue almost every θ∈[0,π). Finally, for ρ=1/4 we prove that the possibility for Eθ to contain an interval is strictly smaller than that for Eθ to have an exact overlap.