The Uncertainty Principle for relatively dense sets and lacunary spectra

Behavior of lacunary series at the natural boundary

We develop a local theory of lacunary Dirichlet series of the form $\sum\limits_{k=1}^{\infty}c_k\exp(-zg(k)), \Re(z)>0$ as $z$ approaches the boundary $i\RR$, under the assumption $g'\to\infty$ and further assumptions on $c_k$. These series occur in many applications in Fourier analysis, infinite order differential operators, number theory and holomorphic dynamics among others. For relatively general series with $c_k=1$, the case we primarily focus on, we obtain blow up rates in measure along the imaginary line and asymptotic information at $z=0$. When sufficient analyticity information on $g…

为什么 Meyer 集会产生衍射？

Why do Meyer sets diffract?

本文研究了 Meyer 集衍射现象的普适性，构造了弱模型集所有子集共享的 Bragg 峰与范数几乎周期集合，并将结果推广到 Meyer 集上的一般加权 Dirac 梳及广义 Eberlein 分解。