Extracting randomness using few independent sources

In this work we give the first deterministic extractors from a constant number of weak sources whose entropy rate is less than 1/2. Specifrically, for every δ > 0 we give an explicit construction for extracting randomness from a constant (depending polynomially on 1/δ) number of distributions over {0, 1}ⁿ, each having min-entropy δn. These extractors output n bits, which are 2^-n close to uniform. This construction uses several results from additive number theory, and in particular a recent one by Bourgain, Katz and Tao [BKT03] and of Konyagin [Kon03]. We also consider the related problem of constructing randomness dispersers. For any constant output length m, our dispersers use a constant number of identical distributions, each with min-entropy Ω(log n) and outputs every possible m-bit string with positive probability. The main tool we use is a variant of the "stepping-up lemma" used in establishing lower bound on the Ramsey number for hyper-graphs (Erdos and Hajnal, [GRS80]).