Finding integral diagonal pairs in a two dimensional $\mathcal{N}$--set

Details Finding integral diagonal pairs in a two dimensional $\mathcal{N}$--set Finding integral diagonal pairs in a two dimensional $\mathcal{N}$--set

2010-07-08

arxiv.org/abs/1007.1441v1

Mathematics

Number Theory

Scientific paper

According to [1] an $n$-dimensional $\mathcal{N}$--set is a compact subset $A$ of $\mathbb{R}^n$ such that for every $x$ in $\mathbb{R}^n$ there is $y$ in $A$ with $y-x$ in $\mathbb{Z}^n$. We prove that every two dimensional $\mathcal{N}$--set $A$ must contain distinct points $x,y$ such that $x-y$ is in $\mathbb{Z}^2$ and $x-y$ is neither horizontal nor vertical. This answers a question of P. Hegarty and M. Nathanson.

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