# 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).

• 2 publications
• 9 publications
research
03/01/2022

### Two Classes of Power Mappings with Boomerang Uniformity 2

Let q be an odd prime power. Let F_1(x)=x^d_1 and F_2(x)=x^d_2 be power ...
research
03/17/2022

### A recursive function coding number theoretic functions

We show that there exists a fixed recursive function e such that for all...
research
10/29/2018

### Location and scale behaviour of the quantiles of a natural exponential family

Let P_0 be a probability on the real line generating a natural exponenti...
research
03/08/2018

### A note on two-colorability of nonuniform hypergraphs

For a hypergraph H, let q(H) denote the expected number of monochromatic...
research
01/11/2023

### A Note on Property Testing of the Binary Rank

Let M be a n× m (0,1)-matrix. We define the s-binary rank, br_s(M), of M...
research
06/07/2020

### Sumsets of Wythoff Sequences, Fibonacci Representation, and Beyond

Let α = (1+√(5))/2 and define the lower and upper Wythoff sequences by a...
research
07/12/2019

### Equiprobable mappings in weighted constraint grammars

We show that MaxEnt is so rich that it can distinguish between any two d...