Abstract
We describe the structure of d-dimensional sets of lattice points, having a small doubling property. Let K be a finite subset of Z d such that dim K = d ⩾ 2 . If | K + K | < ( d + 4 3 ) | K | − 1 6 ( 3 d 2 + 5 d + 8 ) and | K | > 3 ⋅ 4 d , then K lies on d parallel lines. Moreover, for every d-dimensional finite set K ⊆ Z d that lies on d ⩾ 1 parallel lines, if | K + K | < ( d + 2 ) | K | − 1 2 ( d + 1 ) ( d + 2 ) , then K is contained in d parallel arithmetic progressions with the same common difference, having together no more than v = | K + K | − d | K | + 1 2 d ( d + 1 ) 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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