Integral points on strictly convex closed curves

A negative answer is given to Swinnerton-Dyer's question: Is it true that for any > 0 there exists a positive integer n such that for any planar closed strictly convex n-times differentiable curve , when it is blown up a sufficiently large number of times, the number of integral points on the resultant curve will be less than . An example has been constructed when this number for an infinite number is not less than ^1/2, while is infinitely differentiable.Translated from Matematicheskie Zametki, Vol. 21, No. 6, pp. 799–805, June, 1977.The author thanks S. B. Stechkin for attention to the work.