Dynamical spectrum of power-free integers in quadratic number fields and beyond
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| Format: | Preprint |
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2025
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| _version_ | 1866912741795561472 |
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| author | Baake, Michael Luz, Daniel Schindler, Tanja |
| author_facet | Baake, Michael Luz, Daniel Schindler, Tanja |
| contents | 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 $\cB$-free lattice systems in $\RR^d$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_17736 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dynamical spectrum of power-free integers in quadratic number fields and beyond Baake, Michael Luz, Daniel Schindler, Tanja Dynamical Systems Mathematical Physics 37D40, 52C23 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 $\cB$-free lattice systems in $\RR^d$ 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. |
| title | Dynamical spectrum of power-free integers in quadratic number fields and beyond |
| topic | Dynamical Systems Mathematical Physics 37D40, 52C23 |
| url | https://arxiv.org/abs/2502.17736 |