An optimal uncertainty principle in twelve dimensions via modular forms

Abstract

We prove an optimal bound in twelve dimensions for the uncertainty principle of Bourgain, Clozel, and Kahane. Suppose $$f :\mathbb {R}^{12} \rightarrow \mathbb {R}$$ is an integrable function that is not identically zero. Normalize its Fourier transform $$\widehat{f}$$ by $$\widehat{f}(\xi ) = \int _{\mathbb {R}^d} f(x)e^{-2\pi i \langle x, \xi \rangle }\, dx$$ , 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 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.