The structure of d-dimensional sets with small sumset |
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Authors: | Yonutz V Stanchescu |
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Institution: | a Afeka Academic College, Tel Aviv 69107, Israel b The Open University of Israel, Raanana 43107, Israel |
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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 and |K|>3⋅d4, then K lies on d parallel lines. Moreover, for every d-dimensional finite set K⊆Zd that lies on d?1 parallel lines, if , then K is contained in d parallel arithmetic progressions with the same common difference, having together no more than 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|. |
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Keywords: | primary 11P70 secondary 11B75 52C99 |
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