محفوظ في:
| المؤلفون الرئيسيون: | , , , |
|---|---|
| التنسيق: | Preprint |
| منشور في: |
2024
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| الموضوعات: | |
| الوصول للمادة أونلاين: | https://arxiv.org/abs/2404.17536 |
| الوسوم: |
إضافة وسم
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جدول المحتويات:
- We consider the following classical conjecture of Besicovitch: a $1$-dimensional Borel set in the plane with finite Hausdorff $1$-dimensional measure $\mathcal{H}^1$ which has lower density strictly larger than $\frac{1}{2}$ almost everywhere must be countably rectifiable. We improve the best known bound, due to Preiss and Tišer, showing that the statement is indeed true if $\frac{1}{2}$ is replaced by $\frac{7}{10}$ (in fact we improve the Preiss-Tišer bound even for the corresponding statement in general metric spaces). More importantly, we propose a family of variational problems to produce the latter and many other similar bounds and we study several properties of them, paving the way for further improvements.