Integral points on strictly convex closed curves

S. V. Konyagin

Matematičeskie zametki, Tome 21 (1977) no. 6, pp. 799-806.

Résumé

A negative answer is given to Swinnerton–Dyer's question: Is it true that for any $\varepsilon>0$ there exists a positive integer $n$ such that for any planar closed strictly convex $n$-times differentiable curve $\Gamma$, when it is blown up a sufficiently large number $\nu$ of times, the number of integral points on the resultant curve will be less than $\nu^\varepsilon$. An example has been constructed when this number for an infinite number $\nu$ is not less than $\nu^{1/2}$, while $\Gamma$ is infinitely differentiable.