Realizing trees of configurations in thin sets
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908327152189440 |
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| author | Greenleaf, Allan Iosevich, Alex Taylor, Krystal |
| author_facet | Greenleaf, Allan Iosevich, Alex Taylor, Krystal |
| contents | Let $ϕ(x,y)$ be a continuous function, smooth away from the diagonal, such that, for some $α>0$, the associated generalized Radon transforms \begin{equation} \label{Radon} R_t^ϕf(x)=\int_{ϕ(x,y)=t} f(y) ψ(y) dσ_{x,t}(y) \end{equation} map $L^2({\mathbb R}^d) \to L^2_α({\mathbb R}^d)$ for all $t>0$. Let $E$ be a compact subset of ${\mathbb R}^d$ for some $d \ge 2$, and suppose that the Hausdorff dimension of $E$ is $>d-α$. We show that any tree graph $T$ on $k+1$ ($k \ge 1$) vertices is \new{stably} realizable in $E$, in the sense that \new{for each $t$ in some open interval} there exist distinct $x^1, x^2, \dots, x^{k+1} \in E$ %and $t>0$ such that the $ϕ$-distance $ϕ(x^i, x^j)=t$ for all pairs $(i,j)$ corresponding to the edges of $T$.
We extend this result to trees whose edges are prescribed by more complicated point configurations, such as congruence classes of triangles.
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| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_11597 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Realizing trees of configurations in thin sets Greenleaf, Allan Iosevich, Alex Taylor, Krystal Classical Analysis and ODEs Combinatorics Metric Geometry Let $ϕ(x,y)$ be a continuous function, smooth away from the diagonal, such that, for some $α>0$, the associated generalized Radon transforms \begin{equation} \label{Radon} R_t^ϕf(x)=\int_{ϕ(x,y)=t} f(y) ψ(y) dσ_{x,t}(y) \end{equation} map $L^2({\mathbb R}^d) \to L^2_α({\mathbb R}^d)$ for all $t>0$. Let $E$ be a compact subset of ${\mathbb R}^d$ for some $d \ge 2$, and suppose that the Hausdorff dimension of $E$ is $>d-α$. We show that any tree graph $T$ on $k+1$ ($k \ge 1$) vertices is \new{stably} realizable in $E$, in the sense that \new{for each $t$ in some open interval} there exist distinct $x^1, x^2, \dots, x^{k+1} \in E$ %and $t>0$ such that the $ϕ$-distance $ϕ(x^i, x^j)=t$ for all pairs $(i,j)$ corresponding to the edges of $T$. We extend this result to trees whose edges are prescribed by more complicated point configurations, such as congruence classes of triangles. |
| title | Realizing trees of configurations in thin sets |
| topic | Classical Analysis and ODEs Combinatorics Metric Geometry |
| url | https://arxiv.org/abs/2401.11597 |