On arithmetic sums of fractal sets in Rd

摘要

A compact set E subset of Rd is said to be arithmetically thick if there exists a positive integer n so that the n-fold arithmetic sum of E has non-empty interior. We prove the arithmetic thickness of E, if E is uniformly non-flat, in the sense that there exists epsilon 0>0 such that for x is an element of E and 0<r <= diam(E), E boolean AND B(x,r) never stays epsilon 0r-close to a hyperplane in Rd. Moreover, we prove the arithmetic thickness for several classes of fractal sets, including self-similar sets, self-conformal sets in Rd (with d > 2) and self-affine sets in R2 that do not lie in a hyperplane, and certain self-affine sets in Rd (with d > 3) under specific assumptions.

关键词