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 : R 12 → R is an integrable function that is not identically zero. Normalize its Fourier transform f by≥ 0 for |x| ≥ r 1 , and f (ξ) ≥ 0 for |ξ| ≥ 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 to develop a complementary uncertainty principle. This generalization unites the uncertainty principle with the linear programming bound as aspects of a broader theory.

Pair correlation estimates for the zeros of the zeta function via semidefinite programming

“…It is worth mentioning that, in a related uncertainty problem, Cohn and Gonçalves[21] discovered the same kind of instability in low dimensions.…”

A note on Schwartz functions and modular forms

“…After the necessary inequalities are checked, Theorem 1.2 implies that Remark. Henry Cohn has pointed out to the authors that for d = 12 this construction gives the same function as in [3] which proves the optimal upper bound A + (12) = √ 2 for the uncertainty principle of Bourgain, Clozel, and Kahane. He also notes that, after checking some inequalities, this construction would give an upper bound of A + (28) ≤ 2 for d = 28.…”

“…He also notes that, after checking some inequalities, this construction would give an upper bound of A + (28) ≤ 2 for d = 28. Cohn and Gonçalves showed in [3] that A + (28) < 1.99, but they conjecture that there exists an optimal function for d = 28 with non-zero roots at radii 2j + o(1) for j ≥ 2.…”

“…The Schwartz functions constructed by Viazovska, Cohn, Kumar, Miller, Radchenko, and Viazovska, and Cohn and Gonçalves in [11], [5], [6], and [3] showed that there is a surprising and beautiful connection between modular forms and various problems related to sphere packing. It is natural to seek a better understanding of these functions, and to aim for a general framework.…”

Three convolution inequalities on the real line with connections to additive combinatorics

“…It was also shown in [15] that the sharp constant is assumed by a function and that this function has infinitely many double roots. While this question is still open, a sharp form of the inequality in d = 12 dimensions has been established by Cohn & Goncalves [9] using modular forms. There is also at least a philosophical similarity to problems related to the 'unavoidable geometry of probability distributions' [1,10,12,13,25,26].…”