Almost homomorphisms between the Boolean cube and groups of prime order

TL;DR

This paper investigates the probability that an injective function from an n-dimensional Boolean cube to a prime order group approximately preserves addition, showing it diminishes exponentially with the cube's dimension.

Contribution

It establishes a bound on the probability of approximate homomorphisms from Boolean cubes to prime order groups, extending understanding of their structural properties.

Findings

Probability of approximate homomorphism is O(2^{-n/11})

Injective maps rarely preserve addition approximately in high dimensions

Provides bounds on the structure of functions between Boolean cubes and prime groups

Abstract

We show that if f is an injection from an n-dimensional Boolean cube (considered as an additive group) to a group of prime order then the probability that f(x+y)=f(x)+f(y) is O(2^{-n/11}).