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arXiv · 2609.04682

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

Zi-Chao Chi·Jian-Feng Lu·He-Xiang Wang
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Abstract

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

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Zi-Chao Chi, Jian-Feng Lu, He-Xiang Wang. 2026-09-04. Primes are Complete for a Class of $N$-Bernoulli Convolutions Spectral Pair. https://arxiv.org/abs/2609.04682

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