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

It is known that the image in $\mathbb{R}^{2}/\mathbb{Z}^{2}$ of a circle of radius $\rho$ in the plane becomes equidistributed as $\rho\to\infty$. We consider the following sparse version of this phenomenon. Starting from a sequence of radii $\left\{ \rho_{n}\right\} _{n=1}^{\infty}$ which diverges to $\infty$ and an angle $\omega\in\mathbb{R}/\mathbb{Z},$ we consider the projection to $\mathbb{R}^{2}/\mathbb{Z}^{2}$ 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 $\infty$ 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.