Similar point configurations via group actions
Abstract
We prove that for , if the Hausdorff dimension of a compact set is greater than , then, for any given , there exist , , a rotation , and a vector such that for . Such a result on existence of similar -simplices in thin sets had previously been established under a more stringent dimensional threshold in Greenleaf, Iosevich and Mkrtchyan \cite{GIM21}. The argument we are use to prove the main result here was previously employed in Bhowmik and Rakhmonov \cite{BR23} to establish a finite field version. We also show the existence of multi-similarities of arbitrary multiplicity in , show how to extend these results from similarities to arbitrary proper continuous maps, as well as explore a general group-theoretic formulation of this problem in vector spaces over finite fields.
Cite
@article{arxiv.2405.02909,
title = {Similar point configurations via group actions},
author = {P. Bhowmik and A. Greenleaf and A. Iosevich and S. Mkrtchyan and F. Rakhmonov},
journal= {arXiv preprint arXiv:2405.02909},
year = {2024}
}