Characterization on spectra of self-similar measures and the dual spectral set conjecture

A discrete set $Λ$ is called a {\it spectrum} of a Borel probability measure $μ$ if the exponential functions $\{e^{2πi \langleλ, x\rangle}:λ\inΛ\}$ form an orthonormal basis for $L^2(μ)$. In this work, we first give necessary and sufficient conditions for a self-replicating translation set to be a