Lattice points on small arcs
Abstract
We show that for any α∈ (1/2,1) the number of lattice points belonging to an arc of length Rα of the circle of radius R centered at the origin is not uniformly bounded in R, which disproves the corresponding conjecture of Cilleruelo and Granville. We also give certain generalizations of this fact and estimates for the L4-norm of Gauss sums.
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