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Isospectral measures

In recent papers a number of authors have considered Borel probability measures $μ$ in $\br^d$ such that the Hilbert space $L^2(μ)$ has a Fourier basis (orthogonal) of complex exponentials. If $μ$ satisfies this property, the set of frequencies in this set are called a spectrum for $μ$. Here we fix a spectrum, say $Γ$, and we study the possibilities for measures $μ$ having $Γ$ as spectrum.