On A Plane Convex Arc With A Large Number Of Lattice Points

摘要

A convex air on which there are at least M~(long 2/long 3) rational points of the form {u/M, v/M) is constructed. Bibliography: 10 titles.rnIn [1], Jarnik completely solved the problem on the maximum number of integral points on a plane strictly convex arc of a large length l. It turned out that it equals 3(4π)~(-1/3)l~(2/3) + o(l~(2/3)).rnThe corresponding arc is constructed from a given l, and its form varies as l increases.rnIn this connection, the following natural problem statement arises: Given a fixed strictly convex plane arc, compute the number of its rational points having a fixed (large) denominator M.