Almost-nowhere intersection of Cantor sets, and sufficient sampling of their cumulative distribution functions

TL;DR

This paper investigates the properties of Cantor sets and their associated distribution functions, demonstrating that under certain conditions, the sets can be reconstructed from samples of their CDFs, and that their intersections are negligible.

Contribution

It introduces sampling schemes for Cantor set-derived CDFs and proves that two such sets intersect only on a measure-zero set under specific assumptions.

Findings

Cantor sets have almost-nowhere intersection with respect to their invariant measures

Sampling the CDFs allows for reconstruction of the underlying Cantor sets

The intersection of two Cantor sets is negligible under certain conditions

Abstract

Cantor sets are constructed from iteratively removing sections of intervals. This process yields a cumulative distribution function (CDF), constructed from the invariant measure associated with their iterated function systems. Under appropriate assumptions, we identify sampling schemes of such CDFs, meaning that the underlying Cantor set can be reconstructed from sufficiently many samples of its CDF. To this end, we prove that two Cantor sets have almost-nowhere (with respect to their respective invariant measures) intersection.