Lattice points on small arcs

We show that for any $\alpha \in (1\mathrm{/}2,1)$ 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.