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

We study occurrences of point configurations in subsets of $\mathbb{R}^d , d \geq 3$. Given a finite collection of points, a well-known question is: How high does the Hausdorff dimension $\dim_{\mathcal{H}}(E)$ of a compact set $E \subset \mathbb{R}^d$ need to be to ensure that $E$ contains some similar copy of this configuration? We study a related problem, showing that, for $\dim_{\mathcal{H}}(E)$ sufficiently large, $E$ must contain a continuum of point configurations which we call $k$-necklaces of constant gap. Rather than a single geometric shape, a constant-gap $k$-necklace encompasses a family of configurations and may be viewed as a higher dimensional generalization of equilateral triangles and rhombuses in the plane. Our results extend and complement those in [1, 3], where related questions were recently studied.

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Received 8 September 2014

Accepted 5 February 2016

Published 24 July 2017