Convex bodies with a point of curvature do not have Fourier bases

A Weyl type formula for Fourier spectra and frames

We prove qualitative and quantitative results concerning the asymptotic density in dilates of centered convex bodies of the frequency vectors of orthogonal exponential bases and frames associated to bounded domains in Euclidean space.

多维非光滑凸体的差异估计：一个案例研究

Discrepancy estimates for multi-dimensional non-smooth convex bodies: a case study

针对环面上有限点集相对于平移和缩放的非光滑边界凸体的L^2平均差异，在傅里叶变换衰减的各向异性假设下，证明了匹配的上下界，得到速率N^{1 - (d+1)/(d^2+d-1)}，并恢复d=2时的已知指数2/5。

Distance sets corresponding to convex bodies

Suppose that $K \subseteq \RR^d$ is a 0-symmetric convex body which defines the usual norm $$ \Norm{x}_K = \sup\Set{t\ge 0: x \notin tK} $$ on $\RR^d$. Let also $A\subseteq\RR^d$ be a measurable set of positive upper density $ρ$. We show that if the body $K$ is not a polytope, or if it is a polytope with many faces (depending on $ρ$), then the distance set $$ D_K(A) = \Set{\Norm{x-y}_K: x,y\in A} $$ contains all points $t\ge t_0$ for some positive number $t_0$. This was proved by Katznelson and Weiss, by Falconer and Marstrand and by Bourgain in the case where $K$ is the Euclidean ball in any …

高斯曲率非消失光滑曲面上的Fourier框架

Fourier frames on smooth surfaces with nonvanishing Gaussian curvature

本文证明高斯曲率非消失的光滑曲面（如球面）不允许Fourier框架，推广了球冠情形并回答了Kolountzakis和Lai的问题。

关于非滑斜坡：对近期发表于《自然》的一篇关于滚动体论文的若干观察

On the non-slippery slope: Some observations on a recent paper on rolling bodies published in Nature

本文基于高斯定理，对《自然》中构建斜面滚动体的方法进行补充，提供严谨的存在性定理，并关联至海上救援等看似无关的问题。

Confirmation of Matheron's conjecture on the covariogram of a planar convex body

The covariogram g_K of a convex body K in E^d is the function which associates to each x in E^d the volume of the intersection of K with K+x. In 1986 G. Matheron conjectured that for d=2 the covariogram g_K determines K within the class of all planar convex bodies, up to translations and reflections in a point. This problem is equivalent to some problems in stochastic geometry and probability as well as to a particular case of the phase retrieval problem in Fourier analysis. It is also relevant for the inverse problem of determining the atomic structure of a quasicrystal from its X-ray diffrac…