On the Freiman theorem in finite fields

For brevity, we write F2 instead of F {1,...,n} 2 . If J ⊂ I, then there is a natural embedding of F2 in F2 (defined by setting xi = 0 for i ∈ I \ J). In particular, form ≤ n, we consider F2 as a subset of F2 . LetK ≥ 1. Introduce F (K) as the minimal constant such that, for any finite set I and any nonempty A ⊂ F2 with a doubling constant σ(A) ≤ K, there is an affine subspace V ⊂ F2 containing A and of cardinality |V | ≤ F (K)|A|. The assertion claiming that F (K) is finite for anyK ≥ 1 is referred to as the Freiman theorem in F2. Ruzsa [2] proved that F (K) ≤ 2K4 for any K ≥ 1. The upper bounds for F (K) were refined in a series of papers (see [3]–[5]). Green and Tao [6] recently found the asymptotics for the logarithm of F (K) asK → ∞ by establishing the relation F (K) = 2 √ K , K ≥ 2. (1) Moreover, as was noted in [6], F (K) ≥ 22K−O(logK), K ≥ 2. (2) Here and below, log stands for the logarithm to base 2. In the present note, we refine the remainder in (1).