Mathematical Research Letters

Volume 24 (2017)

Number 2

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

Pages: 347 – 362



Allan Greenleaf (Department of Mathematics, University of Rochester, New York, U.S.A.)

Alex Iosevich (Department of Mathematics, University of Rochester, New York, U.S.A.)

Malabika Pramanik (Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada)


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.

Received 8 September 2014

Accepted 5 February 2016

Published 24 July 2017