Power-free integers and related lattice subsets give rise to interesting dynamical systems. They are revisited from a spectral perspective, in the setting of the Halmos--von Neumann theorem. With respect to the natural patch frequency measure, also known as the Mirsky measure, many of these systems have pure-point dynamical spectrum, but trivial topological point spectrum. We calculate the spectra explicitly, in additive notation, and derive their group structure, both for a large class of B-free lattice systems in Rᵈ and for power-free integers in quadratic number fields. Further, in all cases, the eigenfunctions can be given in closed form, via the Fourier--Bohr coefficients on the dense translation orbit of a generic element. Based on a simple argument via Kolmogorov's strong law of large numbers, we show that they are also the eigenfunctions for the unique measure of maximal entropy.
Baake et al. (Mon,) studied this question.