Let V ⊆ [n] be a k-element subset of [n]. The uniform distribution on the 2^k strings from {0, 1}ⁿ that are set to zero outside of V is called an (n, k)-zero-fixing source. An ϵ-extractor for (n, k)-zero-fixing sources is a mapping F: {0, 1}ⁿ → {0, 1}^m, for some m, such that F(X) is ϵ-close in statistical distance to the uniform distribution on {0, 1}^m for every (n, k)-zero-fixing source X. Zero-fixing sources were introduced by Cohen and Shinkar in [7] in connection with the previously studied extractors for bit-fixing sources. They constructed, for every μ > 0, an efficiently computable extractor that extracts a positive fraction of entropy, i.e., Ω(k) bits, from (n, k)-zero-fixing sources where k ≥ (log log n)^2+μ.

In this paper we present two different constructions of extractors for zero-fixing sources that are able to extract a positive fraction of entropy for k substantially smaller than log log n. The first extractor works for k ≥ C log log log n, for some constant C. The second extractor extracts a positive fraction of entropy for k ≥ log⁽ⁱ⁾n for any fixed i ∈ ℕ, where log⁽ⁱ⁾ denotes i-times iterated logarithm. The fraction of extracted entropy decreases with i. The first extractor is a function computable in polynomial time in n; the second one is computable in polynomial time in n when k ≤ α log log n/log log log n, where α is a positive constant.

Our results can also be viewed as lower bounds on some Ramsey-type properties. The main difference between the problems about extractors studied here and the standard Ramsey theory is that we study colorings of all subsets of size up to k while in Ramsey theory the sizes are fixed to k. However it is easy to derive results also for coloring of subsets of sizes equal to k. In Corollary 3.1 of Theorem 5.1 we show that for every l ∈ ℕ there exists β < 1 such that for every k and n, n ≤ exp^l (k), there exists a 2-coloring of k-tuples of elements of [n], \(\psi :\left({\matrix{{[n]} \cr k \cr}} \right) \to \left\{{- 1,1} \right\}\) such that for every V ⊆ [n], |V| = 2k, we have \(\left| {\sum\nolimits_{X \subseteq V,\left| X \right| = k} {\psi (X)}} \right| \le {\beta ^k}\left({\matrix{{2k} \cr k \cr}} \right)\) (Corollary 3.1 is more general — the number of colors may be more than 2).

令V ⊆ [ n ] 是 [ n ] 的k个元素子集。来自 {0, 1} ⁿ且在V之外设置为零的 2 ^k个字符串上的均匀分布称为 ( n, k )-零固定源。( n, k ) 定零源的ϵ -extractor 是一个映射F : {0, 1} ⁿ → {0, 1} ^m，对于某些m，使得F ( X ) 在统计上是ϵ -close对于^每个(n, k )-零固定源X。Cohen 和 Shinkar 在 [7] 中介绍了与先前研究的用于位固定源的提取器相关的零固定源。他们为每个μ > 0 构建了一个高效可计算的提取器，该提取器从 ( n, k ) 的零固定源中提取熵的正分数，即Ω ( k ) 位，其中k ≥ (log log n ) ^2+微米。

在本文中，我们提出了两种不同的零固定源提取器结构，它们能够在k远小于 log log n的情况下提取熵的正分数。第一个提取器适用于k ≥ C log log log n，对于某个常数C。对于任何固定的i ∈ ℕ，第二个提取器提取k ≥ log ^{( i )} n的熵的正分数，其中 log ^{( i )}表示i次迭代对数。提取熵的分数随着i减小. 第一个提取器是一个可在多项式时间内以n计算的函数；当k ≤ α log log n /log log log n时，第二个可在多项式时间内以n计算，其中α是一个正常数。

我们的结果也可以看作是一些 Ramsey 类型属性的下界。这里研究的提取器问题与标准 Ramsey 理论之间的主要区别在于，我们研究了大小为k的所有子集的着色，而在 Ramsey 理论中，大小固定为k。然而，对于大小等于k​​的子集的着色，也很容易得出结果。在定理 5.1 的推论 3.1 中，我们证明对于每个l ∈ ℕ 存在β < 1 使得对于每个k和n，n ≤ exp ^l ( k )，存在[ n元素的k元组的 2 着色],\(\psi :\left({\matrix{{[n]} \cr k \cr}} \right) \to \left\{{- 1,1} \right\}\)使得对于每个V ⊆ [ n ], | 五| = 2 k，我们有\(\left| {\sum\nolimits_{X \subseteq V,\left| X \right| = k} {\psi (X)}} \right| \le {\beta ^k }\left({\matrix{{2k} \cr k \cr}} \right)\)（推论 3.1 更一般——颜色的数量可能超过 2）。