The HRT Conjecture for Symmetric Configurations and Real-Valued Functions
Abstract
The Heil-Ramanathan-Topiwala conjecture asserted that every finite collection of distinct time-frequency shifts of a nonzero square-integrable function is linearly independent. Recent counterexamples with Schwartz windows leave open the classification of configurations and windows for which independence holds. We prove linear independence for every nonzero window in L2(ℝ) and every configuration consisting of finitely many commensurably spaced points on one line together with two distinct points on a parallel line. As a consequence, we obtain independence for every nonzero real-valued window and every four point configuration {(0,0),(0,1),(s,0),(a,b)} with abs≠0. The configurations underlying the known four-point counterexamples can be put in this form by translation and dilation, so they cannot yield dependence with nonzero real-valued windows. The proof combines factorization of Laurent polynomials with product estimates along orbits of irrational rotations. Finally, for a fixed nonzero window and a fixed three-point system, we show that the set of time-frequency parameters producing a shift in its span is compact, has Lebesgue measure zero, has Hausdorff dimension at most one, and is totally disconnected.
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BibTeX
@misc{guan2026conjecture,
title = {{The HRT Conjecture for Symmetric Configurations and Real-Valued Functions}},
author = {Shuang Guan and Kasso A. Okoudjou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.26878},
eprint = {2607.26878},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.26878},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The HRT Conjecture for Symmetric Configurations and Real-Valued Functions
AU - Shuang Guan
AU - Kasso A. Okoudjou
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2607.26878
UR - https://arxiv.org/abs/2607.26878
AB - The Heil-Ramanathan-Topiwala conjecture asserted that every finite collection of distinct time-frequency shifts of a nonzero square-integrable function is linearly independent. Recent counterexamples with Schwartz windows leave open the classification of configurations and windows for which independence holds. We prove linear independence for every nonzero window in L^2(ℝ) and every configuration consisting of finitely many commensurably spaced points on one line together with two distinct points on a parallel line. As a consequence, we obtain independence for every nonzero real-valued window and every four point configuration {(0,0),(0,1),(s,0),(a,b)} with abs≠0. The configurations underlying the known four-point counterexamples can be put in this form by translation and dilation, so they cannot yield dependence with nonzero real-valued windows. The proof combines factorization of Laurent polynomials with product estimates along orbits of irrational rotations. Finally, for a fixed nonzero window and a fixed three-point system, we show that the set of time-frequency parameters producing a shift in its span is compact, has Lebesgue measure zero, has Hausdorff dimension at most one, and is totally disconnected.
ER -