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.
"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