Self-A ne Tiles

A self a ne tile in R is a set T of positive Lebesgue measure satisfying A T d D T d where A is an expanding n n real matrix with j det A j m an integer and D fd dmg R n a set of m digits Self a ne tiles arise in many contexts including radix expansions fractal geometry and the construction of compactly sup ported orthonormal wavelet bases of L R They are also studied as interesting tiles In this article we survey the fundamental properties of self a ne tiles We examine necessary and su cient conditions for digit sets D to give rise to self a ne tiles A special class of self a ne tiles is the integeral self a ne tiles in which A is an integer matrix and D Z We study the tiling properties and the measures of integeral self a ne tiles We also compute the Hausdor dimensions of the boundaries of integeral self a ne tiles Introduction Let A be an expanding matrix in Mn R that is one with all eigenvalues j ij and suppose that jdet A j m for some integer m Let D fd d dmg R n be a nite set of vectors A result of Hutchinson states that there exists a unique nonempty compact set T T A D such that