On a plane convex curve with a large number of lattice points

Let $\gamma$ be a continuous convex curve and let $N_M$ be the number of points belonging to $\gamma$ of the form $(u/M,v/M)$, where $u,v$ are integers.
A smooth curve $\gamma$ such that there exists a sequence $\{M\}$ with the property $N_M>M^{\log2/\log3}$ ($\log2/\log3>0.639$) is constructed. Bibl. – 10 titles.