# On Mappings on the Hypercube with Small Average Stretch

Let A ⊆{0,1}^n be a set of size 2^n-1, and let ϕ{0,1}^n-1→ A be a bijection. We define the average stretch of ϕ as avgStretch(ϕ) = E[ dist(ϕ(x),ϕ(x'))], where the expectation is taken over uniformly random x,x' ∈{0,1}^n-1 that differ in exactly one coordinate. In this paper we continue the line of research studying mappings on the discrete hypercube with small average stretch. We prove the following results. (1) For any set A ⊆{0,1}^n of density 1/2 there exists a bijection ϕ_A {0,1}^n-1→ A such that avgstretch(ϕ_A) = O(√(n)). (2) For n = 3^k let A_ rec-maj = {x ∈{0,1}^n : rec-maj(x) = 1}, where rec-maj : {0,1}^n →{0,1} is the function recursive majority of 3's. There exists a bijection ϕ_ rec-maj{0,1}^n-1→ A_ rec-maj such that avgstretch(ϕ_ rec-maj) = O(1). (3) Let A_ tribes = {x ∈{0,1}^n : tribes(x) = 1}. There exists a bijection ϕ_ tribes{0,1}^n-1→ A_ tribes such that avgstretch(ϕ_ tribes) = O((n)). These results answer the questions raised by Benjamini et al. (FOCS 2014).

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