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 in the plane 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 $\frac{1}{d} Z^2$, 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 the number of spatial Poisson points on \gamma, with intensity $d$, approaches infinity with $d$. The methods include the notion of a generalized affine length of a convex curve, defined by Petrov (2007).