Finding integral diagonal pairs in a two dimensional N--set

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According to [1] an n-dimensional N--set is a compact subset A of Rⁿ such that for every x in Rⁿ there is y in A with y-x in Zⁿ. We prove that every two dimensional N--set A must contain distinct points x,y such that x-y is in Z² and x-y is neither horizontal nor vertical. This answers a question of P. Hegarty and M. Nathanson.