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

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.