标题： 关于与傅里叶基相关的$N$- 伯努利卷积的一个独特性质 显示英文标题

标题： On a distinctive property of Fourier bases associated with $N$- Bernoulli Convolutions

摘要： 关于在 Borel 概率测度 $\mu$ 下的 $\R$ 上的调和分析的一个显著问题是识别所有满足条件的 $t\in\R$，使得 \[\left\{e^{-2\pi i\lambda x}: \lambda\in\Lambda\right\}\quad\text{and}\quad \left\{e^{-2\pi i\lambda x}: \lambda\in t\Lambda\right\}\] 构成空间 $L^2(\mu)$ 的正交基。 目前，这种现象仅在某些奇异测度中被观察到。 它与 Mock 傅里叶级数相对于上述基的收敛性密切相关。 本文中，我们在 $N$-伯努利卷积的框架下应用经典数论解决了该领域中的一个一般猜想和基本问题，这些结果推广了几乎所有已知结果，并得到了一些新的结果。 显示英文摘要

摘要： A distinctive problem of harmonic analysis on $\R$ with respect to a Borel probability measure $\mu$ is identifying all $t\in\R$ such that both \[\left\{e^{-2\pi i\lambda x}: \lambda\in\Lambda\right\}\quad\text{and}\quad \left\{e^{-2\pi i\lambda x}: \lambda\in t\Lambda\right\}\] form orthonormal bases of the space $L^2(\mu)$. Currently, this phenomenon has been observed only in certain singular measures. It is deeply connected to the convergence of Mock Fourier series with respect to the aforementioned bases. In this paper, we apply classical number theory to solve the general conjecture and basic problems in this field within the setting of $N$-Bernoulli convolutions, which extend almost all known results and give some new ones.