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An optimal uncertainty principle in twelve dimensions via modular forms

Henry Cohn,Felipe Gonçalves

Published 2017 in Inventiones Mathematicae

ABSTRACT

We prove an optimal bound in twelve dimensions for the uncertainty principle of Bourgain, Clozel, and Kahane. Suppose $$f :\mathbb {R}^{12} \rightarrow \mathbb {R}$$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.

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