An optimal uncertainty principle in twelve dimensions via modular forms

• 成果类型: Article
• 署名作者: Cohn, Henry; Goncalves, Felipe
• 署名单位: Microsoft; University of Alberta; University of Bonn
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-019-00875-4
• 发表日期: 2019
• 页码: 799-831
• 关键词: sphere packing problem; upper-bounds

摘要

We prove an optimal bound in twelve dimensions for the uncertainty principle of Bourgain, Clozel, and Kahane. Suppose f : R-12 -> R is an integrable function that is not identically zero. Normalize its Fourier transform (f) over cap by (f) over cap(xi) = integral(Rd) f(x)e(-2 pi i < x,xi >) dx, and suppose (f) over cap is real-valued and integrable. We show that if f (0) <= 0, (f) over cap (0) <= 0, f (x) = 0 for vertical bar x vertical bar >= r(1), and (f) over cap(xi) = 0 for vertical bar xi vertical bar >= r(2), then r(1)r(2) >= 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 (f) over cap to develop a complementary uncertainty principle. This generalization unites the uncertainty principle with the linear programming bound as aspects of a broader theory.