The intersection of typical Besicovitch sets with lines

A Quantification of a Besicovitch Nonlinear Projection Theorem via Multiscale Analysis

The Besicovitch projection theorem states that if a subset $E$ of the plane has finite length in the sense of Hausdorff measure and is purely unrectifiable (so its intersection with any Lipschitz graph has zero length), then almost every orthogonal projection of $E$ to a line will have zero measure. In other words, the Favard length of a purely unrectifiable $1$-set vanishes. In this article, we show that when linear projections are replaced by certain nonlinear projections called curve projections, this result remains true. In fact, we go further and use multiscale analysis to prove a quantit…

Bohr and Besicovitch almost periodic discrete sets and quasicrystals

A discrete set $A$ in the Euclidian space is almost periodic if the measure with the unite masses at points of the set is almost periodic in the weak sense. We investigate properties of such sets in the case when $A-A$ is discrete. In particular, if $A$ is a Bohr almost periodic set, we prove that $A$ is a union of a finite number of translates of a certain full--rank lattice. If $A$ is a Besicovitch almost periodic set, then there exists a full-rank lattice such that in most cases a nonempty intersection of its translate with $A$ is large.

Sets with large intersection properties in metric spaces

In this work we reproduce the characterization of $\Gg^s$-sets from the euclidean setting [J. London Math. Soc. 49:267-280,1994] to more general metric spaces. These sets have Hausdorff dimension at least $s$ and are closed by countable intersections, which is particularly useful to estimate the dimension of the so called sets of $α$-approximable points (that typically appear in Diophantine approximations).

可良好与恶劣逼近集合及快速获胜策略

Well and badly approximable sets, and rapid winning

该研究引入Ψ-快速博弈确定τ-逼近数集与非齐次恶劣逼近数集交集的豪斯多夫维数，得到了τ≥1时该交集的精确Jarník–Besicovitch维数。

一类积分几何测度的可求长性及其应用

Rectifiability of a class of integralgeometric measures and applications

本文解决了 Federer 提出的指数 p > 1 积分几何测度可求长性问题，并将其应用于解析容量与可去性的 Vitushkin 猜想，在多尺度和单尺度框架下取得了新结果。