Hermite polynomials, linear flows on the torus, and an uncertainty principle for roots

We study a recent result of Bourgain, Clozel and Kahane, a version of which states that a sufficiently nice function $f:\mathbb{R} \rightarrow \mathbb{R}$ that coincides with its Fourier transform and vanishes at the origin has a root in the interval $(c, \infty)$, where the optimal $c$ satisfies $0.41 \leq c \leq 0.64$. A similar result holds in higher dimensions. We improve the one-dimensional result to $0.45 \leq c \leq 0.594$, and the lower bound in higher dimensions. We also prove that if an extremizer exists, then it has to have infinitely many double roots. The main ingredient here is a new structure statement about Hermite polynomials which relates their pointwise evaluation to linear flows on the torus, and applies to other families of orthogonal polynomials as well.