Zi-Chao Chi | Searcharxiv

arXiv2026

Primes are Complete for a Class of $N$-Bernoulli Convolutions Spectral Pair

We study the complete number problem for a class of self-similar spectral measures on the real line. For a spectral pair $(μ,Λ)$, a real number $t$ is called complete if $tΛ$ is also a spectrum of $μ$. In this paper we consider the $N$-Bernoulli convolution $μ_{N^r,\mathcal D},\mathcal D=\{0,1,\ldots,N-1\},$ together with its spectrum $Λ_{N^r,N^{r-1}\mathcal D}.$ Our main result establishes that every prime number, apart from certain trivial cases, is complete.

math.CA↗

arXiv2025

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

A distinctive problem of harmonic analysis on $\R$ with respect to a Borel probability measure $μ$ is identifying all $t\in\R$ such that both \[\left\{e^{-2πiλx}: λ\inΛ\right\}\quad\text{and}\quad \left\{e^{-2πiλx}: λ\in tΛ\right\}\] form orthonormal bases of the space $L^2(μ)$. 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.

math.CA↗