The structure of d-dimensional sets with small sumset

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|.