Geodesic
On a~plane convex curve with a~large number of lattice points
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 23, Tome 357 (2008), pp. 22-32

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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. 
@article{ZNSL_2008_357_a1,
     author = {E. P. Golubeva},
     title = {On a~plane convex curve with a~large number of lattice points},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {22--32},
     publisher = {mathdoc},
     volume = {357},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2008_357_a1/}
}
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E. P. Golubeva. On a~plane convex curve with a~large number of lattice points. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 23, Tome 357 (2008), pp. 22-32. http://geodesic.mathdoc.fr/item/ZNSL_2008_357_a1/