An optimal uncertainty principle in twelve dimensions via modular forms

TL;DR

This paper establishes a sharp uncertainty principle bound in twelve dimensions using modular forms, connecting it with linear programming bounds and extending the theory to broader contexts.

Contribution

The paper proves an optimal uncertainty principle bound in twelve dimensions using modular form techniques and links it to linear programming bounds, a novel approach in this context.

Findings

Proves a sharp bound of r1*r2 ≥ 2 in 12D for the uncertainty principle.

Constructs a function attaining the bound using Viazovska's modular forms.

Connects the uncertainty principle with linear programming bounds, broadening the theoretical framework.

Abstract

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