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 $widehat{f}(xi) = int_{mathbb{R}^d} f(x)e^{-2pi i langle x, xirangle}, dx$, 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 $widehat{f}(xi) ge 0$ 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 Viazovskas 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.