Magnified curves on a flat torus, determination of almost periodic functions, and the Riemann-Lebesgue lemma

Abstract:

Simple arguments, based on the Riemann-Lebesgue Lemma, are given to show that for a large class of curves $\gamma$ in ${{\mathbf {R}}^n}$, any almost periodic function is determined by its restriction to large dilates of $\gamma$. Specializing to periodic functions, this means that magnified images of $\gamma$ on a flat torus tend to uniformly dense scribbles.