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arXiv · 1604.01462

Multidimensional lower density versions of Plünnecke's inequality

Kamil Bulinski
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Abstract

We investigate the lower asymptotic density of sumsets in $\mathbb{N}^2$ by proving certain Plünnecke type inequalities for various notions of lower density in $\mathbb{N}^2$. More specifically, we introduce a notion of lower tableaux density in $\mathbb{N}^2$ which involves averaging over convex tableaux-shaped regions in $\mathbb{N}^2$ which contain the origin. This generalizes the well known Plünnecke type inequality for the lower asymptotic density of sumsets in $\mathbb{N}$. We also provide a conjectural Plünnecke inequality for the more basic notion of lower rectangular asymtpotic density in $\mathbb{N}^2$ and prove certain partial results.

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Kamil Bulinski. 2017-03-09. Multidimensional lower density versions of Plünnecke's inequality. https://arxiv.org/abs/1604.01462

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