In this paper the following improvement on Freiman's theorem on set addition is obtained (see Theorems 1 and 2 in Section 1).
Let be a finite set such that . Then A is contained in a proper d-dimensional progression P, where and .
Earlier bounds involved exponential dependence in α in the second estimate. Our argument combines I. Ruzsa's method, which we improve in several places, as well as Y. Bilu's proof of Freiman's theorem.
Mei-Chu Chang. "A polynomial bound in Freiman's theorem." Duke Math. J. 113 (3) 399 - 419, 15 June 2002. https://doi.org/10.1215/S0012-7094-02-11331-3