An explicit two-source extractor with min-entropy rate near 4/9

TL;DR

This paper improves the analysis of Bourgain's two-source extractor, achieving extraction from sources with min-entropy rate near 4/9, with exponentially small error, using advanced Fourier analysis and incidence geometry techniques.

Contribution

It provides a refined analysis of Bourgain's extractor, lowering the min-entropy rate threshold to near 4/9 and demonstrating exponential error decay, contrasting with prior methods.

Findings

Extractor works for min-entropy rate near 4/9

Achieves exponentially small error

Builds on recent incidence geometry advances

Abstract

In 2005 Bourgain gave the first explicit construction of a two-source extractor family with min-entropy rate less than $1/2$. His approach combined Fourier analysis with innovative but inefficient tools from arithmetic combinatorics and yielded an unspecified min-entropy rate 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 rate. 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 rate near $\frac{21}{44} =.477\ldots$, moreover we construct a variant of this extractor which we show extracts from sources with min-entropy rate near $4/9 $ = $.444\ldots$. While this min-entropy rate is inferior to Chattopadhyay and Zuckerman's construction, our extractors have the advantage of exponential small error which is important in some applications. 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 Koll\'ar's generalization of the Guth-Katz incidence theorem.