Abstract

Abstract. We determine the constructive dimension of points in random translates of the Cantor set. The Cantor set “cancels randomness ” in the sense that some of its members, when added to Martin-Löf random reals, identify a point with lower constructive dimension than the random itself. In particular, we find the Hausdorff dimension of the set of points in a Cantor set translate with a given constructive dimension. 1. Fractals and randoms We explore an essential interaction between algorithmic randomness, classical fractal geometry, and additive number theory. In this paper, we consider the dimension of the intersection of a given set with a translate of another given set. We shall concern ourselves not only with classical Hausdorff measures and dimension but also the effective analogs of these concepts. More specifically, let C denote the standard middle third Cantor set [6, 18], and for each number α let (1.1) E=α = {x: cdimH{x} = α}

Citations