A pinned Mattila-Sj\"{o}lin type theorem for product sets

TL;DR

This paper extends a theorem about the size of edge length sets in product sets, showing that under certain thickness conditions, these sets contain interior points for configurations related to finite trees.

Contribution

It generalizes McDonald and Taylor's result from Cantor sets on the real line to compact sets in higher-dimensional Euclidean spaces using the concept of thickness.

Findings

Edge length sets have non-empty interior under thickness condition

Generalization from one-dimensional Cantor sets to higher dimensions

Applicable to finite tree configurations in product sets

Abstract

We generalize a result of McDonald and Taylor which concerns the size of the tuples of edge lengths in the set $C_1 \times C_2$ utilizing the notion of thickness. Specifically, we show that $C_1, C_2 \subset \mathbb{R}^d$ compact sets with thickness satisfying $\tau(C_1) \tau(C_2) >1$, then the edge lengths in $C_1 \times C_2$ corresponding to any pinned finite tree configuration has non-empty interior. Originally proven for Cantor sets on the real line by McDonald and Taylor, we use the notion of thickness introduced by Falconer and Yavicoli which allows us to generalize the result of McDonald and Taylor to compact sets in $\mathbb{R}^d$.