Quasihomomorphisms from the integers into Hamming metrics

TL;DR

This paper proves that functions from integers to rational vectors that approximately preserve addition under Hamming distance are close to true homomorphisms, answering a question by Kazhdan and Ziegler.

Contribution

It establishes a bound on how close a quasihomomorphism is to an actual homomorphism, independent of the dimension or specific function.

Findings

Any c-quasihomomorphism is within a bounded distance to a true homomorphism.

The bound depends only on c, not on the dimension n or the specific function.

Provides a positive answer to a special case of a question by Kazhdan and Ziegler.

Abstract

A function $f: \mathbb{Z} \to \mathbb{Q}^n$ is a $c$-quasihomomorphism if the Hamming distance between $f(x+y)$ and $f(x)+f(y)$ is at most $c$ for all $x,y \in \mathbb{Z}$. We show that any $c$-quasihomomorphism has distance at most some constant $C(c)$ to an actual group homomorphism; here $C(c)$ depends only on $c$ and not on $n$ or $f$. This gives a positive answer to a special case of a question posed by Kazhdan and Ziegler.