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:R→R that coincides with its Fourier transform and vanishes at the origin has a root in the interval (c,∞), where the optimal c satisfies 0.41≤c≤0.64. A similar result holds in higher dimensions. We improve the one-dimensional result to 0.45≤c≤0.594, and the lower bound in higher dimensions. We also prove that extremizers exist, and have infinitely many double roots. With this purpose in mind, we establish 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.