Infinite constant gap length trees in products of thick Cantor sets

TL;DR

This paper demonstrates that products of sufficiently thick Cantor sets can generate infinite trees in the plane with a constant gap, and the set of such gaps has non-empty interior, extending previous work on fractal patterns.

Contribution

It introduces the construction of infinite trees with constant gaps in products of thick Cantor sets, allowing countably infinite trees and analyzing the set of possible gaps.

Findings

Products of thick Cantor sets generate trees with constant gaps

The set of possible gap lengths has non-empty interior

The work extends previous results to countably infinite trees

Abstract

We show that products of sufficiently thick Cantor sets generate trees in the plane with constant distance between adjacent vertices. Moreover, we prove that the set of choices for this distance has non-empty interior. We allow our trees to be countably infinite, which further distinguishes this work from previous results on patterns in fractal sets. This builds on the authors' previous work on graphs and distance sets of products of Cantor sets of sufficient Newhouse thickness.