Extractors for Small Zero-Fixing Sources

Let V ⊆ [n] be a k-element subset of [n]. The uniform distribution on the 2k strings from {0, 1}n that are set to zero outside is called an (n, k)-zero-fixing source. An ϵ-extractor for sources mapping F: → 1}m, some m, such F(X) ϵ-close in statistical distance 1}m every source X. Zero-fixing were introduced by Cohen and Shinkar [7] connection with previously studied extractors bit-fixing sources. They constructed, μ > 0, efficiently computable extractor extracts positive fraction entropy, i.e., Ω(k) bits, where k ≥ (log log n)2+μ. In this paper we present two different constructions zero-fixing able extract entropy substantially smaller than n. first works C n, constant C. second log(i)n any fixed i ∈ ℕ, log(i) denotes i-times iterated logarithm. extracted decreases i. function polynomial time n; one n when ≤ α n/log constant. Our results can also viewed as lower bounds Ramsey-type properties. main difference between problems about here standard Ramsey theory study colorings all subsets size up while sizes k. However it easy derive coloring equal Corollary 3.1 Theorem 5.1 show l ℕ there exists β < 1 expl (k), 2-coloring k-tuples elements [n], $$\psi :\left({\matrix{{[n]} \cr \cr}} \right) \to \left\{{- 1,1} \right\}$$ |V| = 2k, have $$\left| {\sum\nolimits_{X \subseteq V,\left| X \right| k} {\psi (X)}} \le {\beta ^k}\left({\matrix{{2k} \right)$$ (Corollary more general — number colors may 2).