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

Abstract

It is known that the image in ℝ2/ℤ2 of a circle of radius ρ in the plane becomes equidistributed as ρ → ∞. We consider the following sparse version of this phenomenon. Starting from a sequence of radii {ρn} ∞n=1 which diverges to ∞ and an angle ω ∈ ℝ/ℤ, we consider the projection to ℝ2/ℤ2 of the n’th roots of unity rotated by angle ω and dilated by a factor of ρn. We prove that if ρn is bounded polynomially in n, then the image of these sparse collections becomes equidistributed, and moreover, if ρngrows arbitrarily fast, then we show that equidistribution holds for almost all ω. Interestingly, we found that for any angle there is a sequence of radii growing to ∞ faster than any polynomial for which equidistribution fails dramatically. In greater generality, we prove this type of results for dilations of varying analytic curves in ℝd. A novel component of the proof is the use of the theory of o-minimal structures to control exponential sums.