Spectral measures associated with the factorization of the Lebesgue measure on a set via convolution (Q485241)

Let \(Q\) be a fundamental domain of some full-rank lattice in \({\mathbb R}^d\) and let \(\mu\) and \(\nu\) be two positive Borel measures on \({\mathbb R}^d\) such that the convolution \(\mu*\nu\) is a multiple of \(\chi_Q\). The authors consider the problem as to whether or not both measures must be spectral and they show that this is true when \(Q=[0,1]^d\). This theorem yields a large class of examples of spectral measures which are either absolutely continuous, singularly continuous or purely discrete spectral measures. In addition, they propose a generalized Fuglede's conjecture for spectral measures on \({\mathbb R}^1\) and they show that it implies the classical Fuglede's conjecture on \({\mathbb R}^1\).