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- W4310688007 abstract "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 $rhotoinfty$. 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 $omegainmathbb{R}/mathbb{Z},$ 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." @default.
- W4310688007 created "2022-12-15" @default.
- W4310688007 creator A5003325916 @default.
- W4310688007 date "2020-03-09" @default.
- W4310688007 modified "2023-10-03" @default.
- W4310688007 title "On the image in the torus of sparse points on dilating analytic curves" @default.
- W4310688007 doi "https://doi.org/10.48550/arxiv.2003.04112" @default.
- W4310688007 hasPublicationYear "2020" @default.
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