On the image in the torus of sparse points on dilating analytic curves

It is known that the image in ${\mathbb{R}}^2\mathrm{/}{\mathbb{Z}}^2$ of a circle of radius $\rho$ in the plane becomes equidistributed as $\rho \to \mathrm{\infty }$. We consider the following sparse version of this phenomenon. Starting from a sequence of radii ${\left\{{\rho }_n\right\}}_{n=1}^{\mathrm{\infty }}$ which diverges to $\mathrm{\infty }$ and an angle $\omega \in \mathbb{R}\mathrm{/}\mathbb{Z},$ we consider the projection to ${\mathbb{R}}^2\mathrm{/}{\mathbb{Z}}^2$ of the $n$’th roots of unity rotated by angle $\omega$ and dilated by a factor of ${\rho }_n$. We prove that if ${\rho }_n$ is bounded polynomially in $n$, then the image of these sparse collections becomes equidistributed, and moreover, if ${\rho }_n$ grows arbitrarily fast, then we show that equidistribution holds for almost all $\omega$. Interestingly, we found that for any angle there is a sequence of radii growing to $\mathrm{\infty }$ faster then any polynomial for which equidistribution fails dramatically. In greater generality, we prove this type of results for dilations of varying analytic curves in ${\mathbb{R}}^d$. A novel component of the proof is the use of the theory of o-minimal structures to control exponential sums.