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Abstract

1980. Proposed by H. A. ShahAli, Tehran, Iran. For every S subset of the plane, let diam(S) = sup{‖x − y‖ : x, y ∈ S}. Let n ≥ 1 be an integer and S1, S2, . . . , Sn subsets of the plane such that ∑n k=1 diam(Sk) < √ 2. Define S = ∪k=1Sk . Prove that there is a translation of S that avoids all points with integer coordinates. That is, prove that there are real numbers r and s such that ((r, s) + S) ∩ (Z × Z) = ∅.