Quasihomomorphisms from the integers into Hamming metrics

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 homomorphis...

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Bibliographic Details
Published in:arXiv.org 2022-04
Main Authors: Draisma, Jan, Eggermont, Rob H, Seynnaeve, Tim, Tairi, Nafie, Ventura, Emanuele
Format: Article
Language:English
Subjects:
Homomorphisms
Online Access:Get full text
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Summary: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.
ISSN:2331-8422