The Minimum Cardinality of a Dependent Finite Gabor System Is Four

Abstract

Recent work produced a linearly dependent system of twelve time--frequency shifts of a Schwartz function, disproving the HRT conjecture. We show that four shifts already suffice, and hence that four is the smallest possible cardinality of a dependent finite Gabor system. More precisely, set $\alpha =\frac{1}{3}+10^{-12}\sqrt{2}$ and $\beta =\frac{1}{3}+10^{-12}\sqrt{3}$. We construct a nonzero complex-valued function $f\in \mathcal{S}(\mathbb{R})$ and $\lambda \ne 0$ such that $\left(I+\frac{1}{2}W(1,0)+\frac{1}{2}W(0,1/2)\right)W(\alpha ,\beta /2)f=\lambda f$, where $W$ denotes the Weyl time--frequency shift. Since every system of at most three shifts of a nonzero $L^2(\mathbb{R})$ function is linearly independent, this gives the sharp cardinality threshold. The construction uses the rank-two Zak bundle naturally associated with the covolume-$1/2$ lattice generated by $(1,0)$ and $(0,1/2)$. At the rational translation $(1/3,1/3)$, the three-step return has a uniformly dominated contracting line. A finite outward-rounded interval certificate proves that this line is topologically trivial. A quantitative perturbation argument carries the dominated line to the explicit algebraic translation above. A winding calculation and a Diophantine cohomological equation then flatten its scalar multiplier, and inverse Zak folding produces the required Schwartz function.