Multidimensional Lower Density Versions of Plünnecke's Inequality

We investigate the lower asymptotic density of sumsets in N2 by proving certain Plünnecke type inequalities for various notions of lower density in N2. More specifically, we introduce a notion of lower tableaux density in N2 which involves averaging over convex tableaux-shaped regions in N2 which contain the origin. This generalizes the well known Plünnecke type inequality for the lower asymptotic density of sumsets in N. We also provide a conjectural Plünnecke inequality for the more basic notion of lower rectangular asymtpotic density in N2 and prove certain partial results.