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Sets of unit vectors with small pairwise sums

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We study the sizes of δ-additive sets of unit vectors in a d-dimensional normed space: the sum of any two vectors has norm at most δ. One-additive sets originate in finding upper bounds of vertex degrees of Steiner Minimum Trees in finite dimensional smooth normed spaces (Z. Füredi, J.C. Lagarias, F. Morgan, 1991). We show that the maximum size of a δ-additive set over all normed spaces of dimension d grows exponentially in d for fixed δ > 2/3, stays bounded for δ < 2/3, and grows linearly at the threshold δ = 2/3. Furthermore, the maximum size of a 2/3-additive set in d-dimensional normed space has the sharp upper bound of d, with the single exception of spaces isometric to three-dimensional l1 space, where there exists a 2/3-additive set of four unit vectors.

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