Translating the Cantor set by a random

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} consist of all real numbers with constructive dimension α. We answer a question posed to us by Doug Hardin by proving the following theorem: if 1− log 2/ log 3 ≤ α ≤ 1 and r is a Martin-Löf random real, then the Hausdorff dimension of (1.2) (C + r) ∩ E=α is α − (1 − log 2/ log 3). From this result we obtain a simple relation between the effective and classical Hausdorff dimensions of (1.2); the difference is exactly 1 minus the dimension of the Cantor set. We conclude that many points in the Cantor set additively cancel randomness. We discuss some of the notions involved in this paper. Intuitively, a real is “random” if it does not inherit any special properties by belonging to an effective null class. We say a number is Martin-Löf random [3, 12] if it “passes” all MartinLöf tests. A Martin-Löf test is a uniformly computably enumerable (c.e.) sequence [3, 16] of open sets {Um}m∈N with λ(Um) ≤ 2, where λ denotes Lebesgue measure [18]. A number x passes such a test if x 6∈ ∩mUm. The Kolmogorov complexity of a string σ, denoted K(σ), is the length (in this paper we will measure length in ternary units) of the shortest program (under a 2000 Mathematics Subject Classification. Primary 68Q30; Secondary 11K55, 28A78.