Trichotomy for the HRT Conjecture for mixed integer configuration
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917348036837376 |
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| author | Oussa, Vignon |
| author_facet | Oussa, Vignon |
| contents | We consider the HRT conjecture in the mixed-integer setting, where a finite configuration in $\R^d\times\R^d$ consists of $N-1$ points in $\Z^d\times\Z^d$ and one point $(α,β)$ outside the lattice. Assuming a linear dependence among the corresponding time-frequency shifts of a nonzero Schwartz function, we apply the Zak transform to obtain a cocycle over translation by $γ=(-α,β)$ on $\T^{2d}$ and study the orbit closure \[ H=\overline{\{nγ\bmod \Z^{2d}:n\in\Z\}}. \] We show that this reduction yields a trichotomy. The dense-orbit case is impossible because a Zak zero propagates to a dense zero set, forcing the Zak transform to vanish identically. The finite-orbit case reduces to a rational configuration, and hence to the lattice case covered by Linnell's theorem. Thus any mixed-integer counterexample for a Schwartz window must occur in the infinite proper case. For that remaining case, we prove that the nonvanishing set of the Zak transform is $H$-saturated, that the averaged logarithmic growth of the modulus cocycle along $H$ exists and vanishes identically, and that the restriction to each nonvanishing $H_0$-coset satisfies a smooth cohomological equation. This yields small-divisor compatibility conditions for the induced translation on $H_0$. We further obtain an arithmetic rigidity condition. These results isolate a collection of necessary dynamical, cohomological, and arithmetic constraints that any mixed-integer counterexample must satisfy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_04613 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Trichotomy for the HRT Conjecture for mixed integer configuration Oussa, Vignon Dynamical Systems Functional Analysis We consider the HRT conjecture in the mixed-integer setting, where a finite configuration in $\R^d\times\R^d$ consists of $N-1$ points in $\Z^d\times\Z^d$ and one point $(α,β)$ outside the lattice. Assuming a linear dependence among the corresponding time-frequency shifts of a nonzero Schwartz function, we apply the Zak transform to obtain a cocycle over translation by $γ=(-α,β)$ on $\T^{2d}$ and study the orbit closure \[ H=\overline{\{nγ\bmod \Z^{2d}:n\in\Z\}}. \] We show that this reduction yields a trichotomy. The dense-orbit case is impossible because a Zak zero propagates to a dense zero set, forcing the Zak transform to vanish identically. The finite-orbit case reduces to a rational configuration, and hence to the lattice case covered by Linnell's theorem. Thus any mixed-integer counterexample for a Schwartz window must occur in the infinite proper case. For that remaining case, we prove that the nonvanishing set of the Zak transform is $H$-saturated, that the averaged logarithmic growth of the modulus cocycle along $H$ exists and vanishes identically, and that the restriction to each nonvanishing $H_0$-coset satisfies a smooth cohomological equation. This yields small-divisor compatibility conditions for the induced translation on $H_0$. We further obtain an arithmetic rigidity condition. These results isolate a collection of necessary dynamical, cohomological, and arithmetic constraints that any mixed-integer counterexample must satisfy. |
| title | Trichotomy for the HRT Conjecture for mixed integer configuration |
| topic | Dynamical Systems Functional Analysis |
| url | https://arxiv.org/abs/2508.04613 |