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Existence of similar point configurations in thin subsets of $${\mathbb {R}}^d$$
- Source :
- Mathematische Zeitschrift. 297:855-865
- Publication Year :
- 2020
- Publisher :
- Springer Science and Business Media LLC, 2020.
-
Abstract
- We prove the existence of similar and multi-similar point configurations (or simplexes) in sets of fractional Hausdorff measure in Euclidean space. These results can be viewed as variants, for thin sets, of theorems for sets of positive density in $\Bbb R^d$ due to Furstenberg, Katznelson and Weiss \cite{FKW90}, Bourgain \cite{B86} and Ziegler \cite{Z06}. Let $d \ge 2$ and $E\subset {\Bbb R}^d$ be a compact set. For $k\ge 1$, define $$\Delta_k(E)=\left\{\left(|x^1-x^2|, \dots, |x^i-x^j|,\dots, |x^k-x^{k+1}|\right): \left\{x^i\right\}_{i=1}^{k+1}\subset E\right\} \subset {\Bbb R}^{k(k+1)/2}, $$ the {\it $(k+1)$-point configuration set} of $E$. For $k\le d$, this is (up to permutations) the set of congruences of $(k+1)$-point configurations in $E$; for $k>d$, it is the edge-length set of $(k+1)$-graphs whose vertices are in $E$. Previous works by a number of authors have found values $s_{k,d}s_{k,d}$, then $\Delta_k(E)$ has positive Lebesgue measure. In this paper we study more refined properties of $\Delta_k(E)$, namely the existence of (exactly) similar or multi--similar configurations. For $r\in\Bbb R,\, r>0$, let $$\Delta_{k}^{r}(E):=\left\{\vec{t}\in \Delta_k\left(E\right): r\vec{t}\in \Delta_k\left(E\right)\right\}\subset \Delta_k\left(E\right).$$ We show that for all $E$ with Hausdorff dimension $>s_{k,d}$, a natural measure $\nu_k$ on $\Delta_k(E)$ and all $r\in\Bbb R_+$, one has $\nu_k\left(\Delta_{k}^{r}\left(E\right)\right)>0$. Thus, there exist many pairs, $\{x^1, x^2, \dots, x^{k+1}\}$ and $\{y^1, y^2, \dots, y^{k+1}\}$, in $E$ which are similar by the scaling factor $r$. We also show the existence of triply-similar and multi-similar configurations.<br />Comment: 14 pages; end of Sec. 4 clarified; additional comment and reference added in Sec. 5
- Subjects :
- Simplex
Lebesgue measure
Euclidean space
General Mathematics
010102 general mathematics
Multiplicity (mathematics)
Congruence relation
01 natural sciences
Combinatorics
Compact space
Natural measure
Mathematics - Classical Analysis and ODEs
Hausdorff dimension
0103 physical sciences
Classical Analysis and ODEs (math.CA)
FOS: Mathematics
010307 mathematical physics
0101 mathematics
Mathematics
Subjects
Details
- ISSN :
- 14321823 and 00255874
- Volume :
- 297
- Database :
- OpenAIRE
- Journal :
- Mathematische Zeitschrift
- Accession number :
- edsair.doi.dedup.....1e52c4a4f939824301cbe4298e4ff247