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We study spectral measures generated by infinite convolution products ofdiscrete measures generated by Hadamard triples, and we present sufficientconditions for the measures to be spectral, generalizing a criterion byStrichartz. We then study the spectral measures generated by randomconvolutions of finite atomic measures and rescaling, where the digits arechosen from a finite collection of digit sets. We show that in dimension one,or in higher dimensions under certain conditions, "almost all" such measuresgenerate spectral measures, or, in the case of complete digit sets,translational tiles. Our proofs are based on the study of self-affine spectralmeasures and tiles generated by Hadamard triples in quasi-product form.

Spectral measures generated by arbitrary and random convolutions