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

Abstract

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. e(m) = 12 (4π2) 1 3m 2 3+O(m 1 3log m). s(n) = 2τ 12 3 2n 3 2+O(nlogn).