十二维最优测不准原理的模形式
An optimal uncertainty principle in twelve dimensions via modular forms
摘要：
我们证明了一个最佳的边界在12个维度的不确定性原则的Bourgain，Clozel，和Kahane。假设$f \colon \mathbb{R}^{12} \to \mathbb{R}$是一个不恒为零的可积函数。用$\widehat{f}(ξ) = \int_{\mathbb{R}^d} f(x)e^{-2πi \langle x, ξ\rangle}\, dx$规范化其傅里叶变换$\widehat{f}$，并假设$\widehat{f}$是实值可积的。我们证明，如果$f(0) \le 0$，$\widehat{f}(0) \le 0$，$f(x) \ge 0$的$|x| \ge r_1$，和$\widehat{f}(ξ) \ge 0$的$|ξ| \ge r_2$，那么$r_1r_2 \ge 2$，这个界限是尖锐的。函数达到上界的构造是基于Viazovska的模形式技术，其最优性来自爱森斯坦级数$E_6$的存在。在任何其他维度上，我们都不知道，甚至不知道有什么明显的界限。我们还开发了一个连接与线性规划界的Cohn和Elkies，这让我们推广的符号模式的$f$和$\widehat{f}$开发一个互补的不确定性原则。这种推广将不确定性原理与线性规划界作为更广泛理论的一个方面。
英文摘要：
We prove an optimal bound in twelve dimensions for the uncertainty principle of Bourgain, Clozel, and Kahane. Suppose $f \colon \mathbb{R}^{12} \to \mathbb{R}$ is an integrable function that is not identically zero. Normalize its Fourier transform $\widehat{f}$ by $\widehat{f}(ξ) = \int_{\mathbb{R}^d} f(x)e^{-2πi \langle x, ξ\rangle}\, dx$, and suppose $\widehat{f}$ is real-valued and integrable. We show that if $f(0) \le 0$, $\widehat{f}(0) \le 0$, $f(x) \ge 0$ for $|x| \ge r_1$, and $\widehat{f}(ξ) \ge 0$ for $|ξ| \ge r_2$, then $r_1r_2 \ge 2$, and this bound is sharp. The construction of a function attaining the bound is based on Viazovska's modular form techniques, and its optimality follows from the existence of the Eisenstein series $E_6$. No sharp bound is known, or even conjectured, in any other dimension. We also develop a connection with the linear programming bound of Cohn and Elkies, which lets us generalize the sign pattern of $f$ and $\widehat{f}$ to develop a complementary uncertainty principle. This generalization unites the uncertainty principle with the linear programming bound as aspects of a broader theory.
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