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.

中文翻译：

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最小熵率接近的显式双源提取器

2005 年，Bourgain 首次明确构建了最小熵率小于 $1/2$ 的双源提取器族。他的方法将傅立叶分析与算术组合学中的创新但效率低下的工具相结合，并产生了一个未指定的最小熵率，其大于 0.499 美元。直到 2015 年 Chattopadhyay 和 Zuckerman 的突破，这基本上仍然是最先进的技术，他们提出了一种替代方法，该方法可以生产具有任意小的最小熵率的提取器。在当前的工作中，我们重新审视了傅立叶分析方法。我们对 Bourgain 的一个提取器进行了改进分析，这表明它实际上从最小熵率接近 $\frac{21}{44} =.477\ldots$ 的源中提取，此外，我们构建了这个提取器的一个变体，我们展示了从最小熵率接近 $4/9 $ = $.444\ldots$ 的源中提取的数据。虽然这个最小熵率不如 Chattopadhyay 和 Zuckerman 的构造，但我们的提取器具有指数小误差的优势，这在某些应用中很重要。这些论点的关键因素是与 Rudnev 和 Shkredov 的有限场抛物面限制理论相关的最新进展。这反过来又依赖于 Rudnev 的点平面关联估计，而后者又依赖于 Koll\'ar 对 Guth-Katz 关联定理的概括。我们的提取器具有指数级小误差的优势，这在某些应用中很重要。这些论点的关键因素是与 Rudnev 和 Shkredov 的有限场抛物面限制理论相关的最新进展。这反过来又依赖于 Rudnev 的点平面关联估计，而后者又依赖于 Koll\'ar 对 Guth-Katz 关联定理的概括。我们的提取器具有指数级小误差的优势，这在某些应用中很重要。这些论点的关键因素是与 Rudnev 和 Shkredov 的有限场抛物面限制理论相关的最新进展。这反过来又依赖于 Rudnev 的点平面关联估计，而后者又依赖于 Koll\'ar 对 Guth-Katz 关联定理的概括。