AI 中文总结
本文证明相关有限Gabor系统的最小基数为4，通过构造含Weyl时频平移的Schwartz函数系统，结合Zak丛等工具否定HRT猜想，给出该基数阈值的严格结果。
AI 中文摘要
近期研究构造了一个由Schwartz函数的12个时频平移构成的线性相关系统，否定了HRT猜想。本文证明4个平移已足够，因此4是相关有限Gabor系统的最小可能基数。更精确地，设α=1/3+10^(-12)√2，β=1/3+10^(-12)√3，我们构造非零复值函数f∈S(R)及λ≠0，使得(I+1/2 W(1,0)+1/2 W(0,1/2))W(α,β/2)f=λf，其中W表示Weyl时频平移。由于非零L²(R)函数的至多3个平移构成的每个系统都是线性无关的，这给出了尖锐的基数阈值。该构造使用与由(1,0)和(0,1/2)生成的余体积1/2格自然关联的秩2 Zak丛，在有理平移(1/3,1/3)处，三步返回具有一致主导收缩线，有限外圆角区间证书证明该线拓扑平凡，定量扰动论证将主导线映射到上述显式代数平移，卷绕计算与Diophantine上同调方程随后展平其标量乘子，逆Zak折叠生成所需的Schwartz函数。
英文摘要
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 $α=\frac13+10^{-12}\sqrt2$ and $β=\frac13+10^{-12}\sqrt3$. We construct a nonzero complex-valued function $f\in\mathcal S(\mathbb R)$ and $λ\ne0$ such that $\left(I+\frac12W(1,0)+\frac12W(0,1/2)\right)W(α,β/2)f=λf$, 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.
