Let be elements of an arbitrary commutative group and let be positive integers. A set of the form
is called a -dimensional generalized arithmetic progression. Its size is defined to be the quantity .
The author proves the following theorem: Let , be finite sets in a torsionfree commutative group with and . Then there are numbers and depending only on such that is contained in a generalized arithmetic progression of dimension at most and of size at most .
This result, in the author’s opinion, is essentially equivalent to a famous theorem of Freiman although it is expressed in different terms and the proof is along completely different lines.