# Translating the Cantor set by a random

Presented by:
J Teutsch Pennsylvania State University
Date:
Thursday 5th July 2012 - 10:00 to 10:30
Venue:
INI Seminar Room 1
Abstract:
I will discuss 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.

More specifically, let $\mathcal{C}$ denote the standard middle third Cantor set, and for each real $\alpha$ let $\mathcal{E}_{\alpha}$ consist of all real numbers with constructive dimension $\alpha$. Our result is the following.

If $1 -\log2/\log3 \leq \alpha \leq 1$ and $r$ is a Martin-Löf random real, then the Hausdorff dimension of the set $(\mathcal{C}+r) \cap \mathcal{E}_{\alpha}$ is $\alpha -(1 -\log 2/\log 3)$.

From this theorem we obtain a simple relation between the effective and classical Hausdorff dimensions of this set; the difference is exactly $1$ minus the dimension of the Cantor set. We conclude that many points in the Cantor set additively cancel randomness.

On the surface, the theorem above describes a connection between algorithmic randomness and classical fractal geometry. Less obvious is its relationship to additive number theory. In 1954, G. G. Lorentz proved the following statement.

There exists a constant $c$ such that for any integer $k$, if $A\subseteq [0, k)$ is a set of integers with ${\left|A\right|} \geq \ell \geq 2$, then there exists a set of integers $B \subseteq (-k,k)$ such that $A + B \supseteq [0, k)$ with ${\left|B\right|} \leq ck\frac{\log \ell}{\ell}$.

Given a Martin-Löf random real $r$, I will show how Lorentz's Lemma can be used to identify a point $x\in\mathcal{C}$ such that the constructive dimension of $x+r$ is close to $1 - \log 2 / \log 3$, which is as small as it can possibly be.

This talk is based on joint work with Randy Dougherty, Jack Lutz, and Dan Mauldin.

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