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

Given , the four corner Cantor set is a self-similar set generated by the iterated function system For let be the orthogonal projection of onto a line with an angle to the -axis. In principle, is a self-similar set having overlaps. In this paper we give a complete characterization on which the projection is totally self-similar. We also study the spectrum of , which turns out that the spectrum achieves its maximum value if and only if is totally self-similar. Furthermore, when is totally self-similar, we calculate its Hausdorff dimension and study the subset which consists of all having a unique coding. In particular, we show that for Lebesgue almost every . Finally, for we prove that the possibility for to contain an interval is strictly smaller than that for to have an exact overlap.