The visible part of plane self-similar sets

Kenneth J. Falconer, Jonathan M. Fraser
2010 arXiv   pre-print
Given a compact subset F of R^2, the visible part V_θ F of F from direction θ is the set of x in F such that the half-line from x in direction θ intersects F only at x. It is suggested that if _H F ≥ 1 then _H V_θ F = 1 for almost all θ, where _H denotes Hausdorff dimension. We confirm this when F is a self-similar set satisfying the convex open set condition and such that the orthogonal projection of F onto every line is an interval. In particular the underlying similarities may involve arbitrary rotations and F need not be connected.
arXiv:1004.5067v1 fatcat:b26i7nizlfcm7p6uhgz6uf2cna