Lattice points on small arcs

TL;DR

The paper proves that the number of lattice points on small arcs of circles of radius R grows unbounded as R increases, refuting a previous conjecture and providing related estimates for Gauss sums.

Contribution

It disproves the conjecture that lattice points on small arcs are uniformly bounded and offers generalizations and Gauss sum estimates.

Findings

Number of lattice points on arcs of length R^α is unbounded for α in (1/2,1)

Disproves Cilleruelo and Granville's conjecture

Provides estimates for the L_4-norm of Gauss sums

Abstract

We show that for any $\alpha\in (1/2,1)$ the number of lattice points belonging to an arc of length $R^{\alpha}$ 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 $L_4$-norm of Gauss sums.