On the maximal number of edges of convex digital polygons included into an m × m-grid

Let e(m) denote the maximal number of edges of a convex digital polygon included into an m × m square area of lattice points and let s(n) denote the minimal (side) size of a square in which a convex digital polygon with n edges can be included. We prove that 12(4π2)13m23+O(m13log m) 2τ1232n32+O(nlogn).