English

Similar point configurations via group actions

Classical Analysis and ODEs 2024-05-07 v1 Combinatorics Metric Geometry

Abstract

We prove that for d2,k2d\ge 2,\, k\ge 2, if the Hausdorff dimension of a compact set ERdE\subset \mathbb{R}^d is greater than d22d1\frac{d^2}{2d-1}, then, for any given r>0r > 0, there exist (x1,,xk+1)Ek+1(x^1, \dots, x^{k+1})\in E^{k+1}, (y1,,yk+1)Ek+1(y^1, \dots, y^{k+1})\in E^{k+1}, a rotation θOd(R)\theta \in \mathrm{O}_d(\mathbb{R}), and a vector aRda \in \mathbb{R}^d such that rxj=θyjarx^j = \theta y^j - a for 1jk+11 \leq j \leq k+1. Such a result on existence of similar kk-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 Rd\R^d, 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.

Keywords

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}
}
R2 v1 2026-06-28T16:17:08.483Z