LAUSR

We prove an optimal bound in twelve dimensions for the uncertainty principle of Bourgain, Clozel, and Kahane. Suppose $$f :\mathbb {R}^{12} \rightarrow \mathbb {R}$$f:R12→R is an integrable function that is not identically zero. Normalize its Fourier transform $$\widehat{f}$$f^ by $$\widehat{f}(\xi ) = \int _{\mathbb {R}^d} f(x)e^{-2\pi i \langle x, \xi \rangle }\, dx$$f^(ξ)=∫Rdf(x)e-2πi⟨x,ξ⟩dx, and suppose $$\widehat{f}$$f^ is real-valued and integrable. We show that if $$f(0) \le 0$$f(0)≤0, $$\widehat{f}(0) \le 0$$f^(0)≤0, $$f(x) \ge 0$$f(x)≥0 for $$|x| \ge r_1$$|x|≥r1, and $$\widehat{f}(\xi ) \ge 0$$f^(ξ)≥0 for $$|\xi | \ge r_2$$|ξ|≥r2, then $$r_1r_2 \ge 2$$r1r2≥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 $$E_6$$E6. 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 $$\widehat{f}$$f^ to develop a complementary uncertainty principle. This generalization unites the uncertainty principle with the linear programming bound as aspects of a broader theory.