Symmetric cantor measure, coin-tossing and sum sets

Abstract

Construct a probability measure μ on the circle by successive removal of middle third intervals with redistributions of the existing mass at the nth stage being determined by probability pn applied uniformly across that level. Assume that the sequence {pn} is bounded away from both 0 and 1. Then, for sufficiently large N, (estimates are given) the Lebesgue measure of any algebraic sum of Borel sets E1, E2,..., EN exceeds the product of the corresponding μ (Ei) α, where α is determined by N and {pn}. It is possible to replace 3 by any integer M ≥ 2 and to work with distinct measures μ1, μ2, ........, μN. This substantially generalizes work of Williamson and the author (for powers of singlecoin coin-tossing measures in the case M = 2) and is motivated by the extension to M = 3. We give also a simple proof of a result of Yin and the author for random variables whose binary digits are determined by coin-tossing.