Similar point configurations via group actions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913341822205952 |
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| author | Bhowmik, P. Greenleaf, A. Iosevich, A. Mkrtchyan, S. Rakhmonov, F. |
| author_facet | Bhowmik, P. Greenleaf, A. Iosevich, A. Mkrtchyan, S. Rakhmonov, F. |
| contents | We prove that for $d\ge 2,\, k\ge 2$, if the Hausdorff dimension of a compact set $E\subset \mathbb{R}^d$ is greater than $\frac{d^2}{2d-1}$, then, for any given $r > 0$, there exist $(x^1, \dots, x^{k+1})\in E^{k+1}$, $(y^1, \dots, y^{k+1})\in E^{k+1}$, a rotation $θ\in \mathrm{O}_d(\mathbb{R})$, and a vector $a \in \mathbb{R}^d$ such that $rx^j = θy^j - a$ for $1 \leq j \leq k+1$. Such a result on existence of similar $k$-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 $\R^d$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02909 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Similar point configurations via group actions Bhowmik, P. Greenleaf, A. Iosevich, A. Mkrtchyan, S. Rakhmonov, F. Classical Analysis and ODEs Combinatorics Metric Geometry We prove that for $d\ge 2,\, k\ge 2$, if the Hausdorff dimension of a compact set $E\subset \mathbb{R}^d$ is greater than $\frac{d^2}{2d-1}$, then, for any given $r > 0$, there exist $(x^1, \dots, x^{k+1})\in E^{k+1}$, $(y^1, \dots, y^{k+1})\in E^{k+1}$, a rotation $θ\in \mathrm{O}_d(\mathbb{R})$, and a vector $a \in \mathbb{R}^d$ such that $rx^j = θy^j - a$ for $1 \leq j \leq k+1$. Such a result on existence of similar $k$-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 $\R^d$, 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. |
| title | Similar point configurations via group actions |
| topic | Classical Analysis and ODEs Combinatorics Metric Geometry |
| url | https://arxiv.org/abs/2405.02909 |