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The structure of d-dimensional sets with small sumset
Authors:Yonutz V Stanchescu
Institution:a Afeka Academic College, Tel Aviv 69107, Israel
b The Open University of Israel, Raanana 43107, Israel
Abstract:We describe the structure of d-dimensional sets of lattice points, having a small doubling property. Let K be a finite subset of Zd such that dimK=d?2. If View the MathML source and |K|>3⋅d4, then K lies on d parallel lines. Moreover, for every d-dimensional finite set KZd that lies on d?1 parallel lines, if View the MathML source, then K is contained in d parallel arithmetic progressions with the same common difference, having together no more than View the MathML source terms. These best possible results answer a recent question posed by Freiman and cannot be sharpened by reducing the quantity v or by increasing the upper bounds for |K+K|.
Keywords:primary  11P70  secondary  11B75  52C99
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