An explicit two-source extractor with min-entropy near 4/9
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In 2005 Bourgain gave the first explicit construction of a two-source extractor family with min-entropy less than 1/2. His approach combined Fourier analysis with innovative but inefficient tools from arithmetic combinatorics and yielded an unspecified min-entropy which was greater than .499. This remained essentially the state of the art until a 2015 breakthrough of Chattopadhyay and Zuckerman in which they gave an alternative approach which produced extractors with arbitrarily small min-entropy. In the current work, we revisit the Fourier analytic approach. We give an improved analysis of one of Bourgain's extractors which shows that it in fact extracts from sources with min-entropy near 21/44 =.477..., moreover we construct a variant of this extractor which we show extracts from sources with min-entropy near 4/9 = .444.... The key ingredient in these arguments is recent progress connected to the restriction theory of the finite field paraboloid by Rudnev and Shkredov. This in turn relies on a Rudnev's point-plane incidence estimate, which in turn relies on Kollár's generalization of the Guth-Katz incidence theorem.
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