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 ${\mu }_{N^r,\mathcal{D}},\mathcal{D}=\{0,1,\dots ,N-1\},$ 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.