The intersection of algorithmically random closed sets and effective dimension

TL;DR

This paper explores the properties of intersections of algorithmically random closed sets, demonstrating how such intersections can be inverted and characterized in terms of effective Hausdorff dimension.

Contribution

It answers a key question about the invertibility of intersections of relatively random closed sets and links these intersections to effective Hausdorff dimension.

Findings

Intersection of relatively random closed sets can be inverted to produce a random closed set.

Identifies Bernoulli measures under which intersections of random closed sets are non-empty.

Characterizes effective Hausdorff dimension via the intersectability of random closed sets.

Abstract

In this article, we study several aspects of the intersections of algorithmically random closed sets. First, we answer a question of Cenzer and Weber, showing that the operation of intersecting relatively random closed sets (with respect to certain underlying measures induced by Bernoulli measures on the space of codes of closed sets), which preserves randomness, can be inverted: a random closed set of the appropriate type can be obtained as the intersection of two relatively random closed sets. We then extend the Cenzer/Weber analysis to the intersection of multiple random closed sets, identifying the Bernoulli measures with respect to which the intersection of relatively random closed sets can be non-empty. We lastly apply our analysis to provide a characterization of the effective Hausdorff dimension of sequences in terms of the degree of intersectability of random closed sets that contain them.