Lebesgue measure of distance sets with regular pins and multi-scale Mizohata-Takeuchi-type estimates
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866910165106688000 |
|---|---|
| author | Liu, Bochen |
| author_facet | Liu, Bochen |
| contents | Suppose $E, F$ are Borel sets in the plane, $\dim_{\mathcal{H}} E>1$, $\dim_{\mathcal{H}} E+\dim_{\mathcal{H}} F>2$, and $F$ has equal Hausdorff and packing dimension. We prove that there exists $y\in F$ such that the pinned distance set $$Δ_y(E):=\{|x-y|:x\in E\}$$ has positive Lebesgue measure. In particular, it settles the regular case of the distance set problem in the plane. The main ingredients of the proof consist of a multi-scale Good-Bad decomposition and a multi-scale Mizohata-Takeuchi-type estimate with arbitrary small power-loss. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_15328 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lebesgue measure of distance sets with regular pins and multi-scale Mizohata-Takeuchi-type estimates Liu, Bochen Classical Analysis and ODEs Analysis of PDEs Combinatorics Suppose $E, F$ are Borel sets in the plane, $\dim_{\mathcal{H}} E>1$, $\dim_{\mathcal{H}} E+\dim_{\mathcal{H}} F>2$, and $F$ has equal Hausdorff and packing dimension. We prove that there exists $y\in F$ such that the pinned distance set $$Δ_y(E):=\{|x-y|:x\in E\}$$ has positive Lebesgue measure. In particular, it settles the regular case of the distance set problem in the plane. The main ingredients of the proof consist of a multi-scale Good-Bad decomposition and a multi-scale Mizohata-Takeuchi-type estimate with arbitrary small power-loss. |
| title | Lebesgue measure of distance sets with regular pins and multi-scale Mizohata-Takeuchi-type estimates |
| topic | Classical Analysis and ODEs Analysis of PDEs Combinatorics |
| url | https://arxiv.org/abs/2603.15328 |