On necklaces inside thin subsets of ${\Bbb R}^d$

TL;DR

This paper investigates the Hausdorff dimension thresholds needed for a set in Euclidean space to contain specific point configurations called k-necklaces with constant gaps, extending previous work on similar geometric patterns.

Contribution

It introduces new results establishing that sufficiently large Hausdorff dimension guarantees the presence of k-necklaces, generalizing known configurations like equilateral triangles and rhombuses.

Findings

Sets with large Hausdorff dimension contain many k-necklaces of constant gap

Results extend previous work on geometric configurations in fractal sets

Provides conditions under which specific point configurations must exist

Abstract

We study similarity classes of point configurations in $\R^d$. Given a finite collection of points, a well-known question is: How high does the Hausdorff dimension $\hd(E)$ of a compact set $E \subset {\Bbb R}^d$, $d \ge 2$, need to be to ensure that $E$ contains some similar copy of this configuration? We prove results for a related problem, showing that for $\hd(D)$ sufficiently large, $E$ must contain many point configurations that we call $k$-necklaces of constant gap, generalizing equilateral triangles and rhombuses in higher dimensions. Our results extend and complement those in \cite{CLP14,BIT14}, where related questions were recently studied.