An optimal uncertainty principle in twelve dimensions via modular forms

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 $\widehat{f}(\xi) = \int_{$\mathbb{R}$^d} f(x)e^{-2\pi i \langle x, \xi\rangle}, dx$,andsuppose$\widehat{f}$ 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$\widehat{f}(\xi) \ge 0$for$|\xi| \ge r_2$,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.