Translating the Cantor set by a random
Randall Dougherty, Jack Lutz, R. Daniel Mauldin, and Jason Teutsch
TL;DR
This paper investigates how translating the Cantor set by a random element affects the constructive dimension of points, revealing that some members reduce the dimension of the sum with random reals.
Contribution
It provides a precise characterization of the constructive dimension of points in random translates of the Cantor set, linking randomness and fractal dimensions.
Findings
Identifies how Cantor set translations influence constructive dimension.
Determines the Hausdorff dimension of points with specific constructive dimensions.
Shows that some Cantor set members lower the dimension of sums with random reals.
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-Lof 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.
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