CONVEX CURVES AND A POISSON IMITATION OF LATTICES

We solve a randomized version of the following open question: is there a strictly convex, bounded curve gamma subset of R-2 such that the number of rational points on gamma, with denominator n, approaches infinity with n? Although this natural problem appears to be out of reach using current methods, we consider a probabilistic analogue using a spatial Poisson process that simulates the refined rational lattice (1/d)Z(2), which we call M-d, for each natural number d. The main result here is that with probability 1 there exists a strictly convex, bounded curve gamma such that |gamma boolean AND M-d| -> +infinity, as d tends to infinity. The methods include the notion of a generalized affine length of a convex curve as defined by F. V. Petrov [Estimates for the number of rational points on convex curves and surfaces. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 344 (2007), 174-189; Engl. transl. J. Math. Sci. 147(6) (2007), 7218-7226].