Dynamical spectrum of power-free integers in quadratic number fields and beyond

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 $ \mathcal{B}$ -free lattice systems in $ \mathbb{R}^d$ 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 of generic elements and their translates, which form a subset of full Mirsky measure. Based on a simple argument via Kolmogorov's strong law of large numbers, we show how the Fourier–Bohr coefficients also provide the eigenfunctions for the unique measure of maximal entropy, and that we get phase consistency for both measures.