A solution to a problem of Cassels and Diophantine properties of cubic numbers

Simultaneous Diophantine approximation - logarithmic improvements

This paper is devoted to the study of a problem of Cassels in multiplicative Diophantine approximation which involves minimising values of a product of affine linear forms computed at integral points. It was previously known that values of this product become arbitrary close to zero, and we establish that, in fact, they approximate zero with an explicit rate. Our approach is based on investigating quantitative density of orbits of higher-rank abelian groups.

Diophantine approximations on fractals

We exploit dynamical properties of diagonal actions to derive results in Diophantine approximations. In particular, we prove that the continued fraction expansion of almost any point on the middle third Cantor set (with respect to the natural measure) contains all finite patterns (hence is well approximable). Similarly, we show that for a variety of fractals in [0,1]^2, possessing some symmetry, almost any point is not Dirichlet improvable (hence is well approximable) and has property C (after Cassels). We then settle by similar methods a conjecture of M. Boshernitzan saying that there are no …

由算术序列驱动的圆覆盖：一种渗透方法用于迪奥菲安逼近和分形交集

Circle coverings driven by arithmetic sequences: a percolation approach to Diophantine approximation and fractal intersections

研究通过缩小区间覆盖[0,1)的问题，揭示了在特定序列下覆盖半径的最优性，并扩展了对分形交集的尺寸计算。

三次散焦非线性薛定谔方程的概周期解

Almost Periodic Solutions of The Cubic Defocusing Nonlinear Schrödinger Equation

本文结合正、逆谱理论，证明满足丢番图频率条件的小解析拟周期初值下，三次散焦NLS方程存在时空概周期且唯一的局部有界解，并解决了相关谱理论问题。