Automatic continuity of Polynomial maps and cocycles

TL;DR

This paper proves that Lebesgue measurable functions on real numbers vanishing under certain difference operators are polynomials, and extends this to the automatic continuity of polynomial maps and cocycles between locally compact groups.

Contribution

It establishes the automatic continuity of Haar measurable polynomial maps and cocycles, generalizing classical results to a broader group-theoretic context.

Findings

Lebesgue measurable functions vanishing under difference operators are polynomials

Haar measurable polynomial maps between locally compact groups are continuous

Automatic continuity of cocycles is demonstrated

Abstract

Classical theorems from the early 20th century state that any Haar measurable homomorphism between locally compact groups is continuous. In particular, any Lebesgue-measurable homomorphism $\phi:\mathbb{R} \to \mathbb{R}$ is of the form $\phi(x)=ax$ for some $a \in \mathbb{R}$. In this short note, we prove that any Lebesgue measurable function $\phi:\mathbb{R} \to \mathbb{R}$ that vanishes under any $d+1$ ``difference operators'' is a polynomial of degree at most $d$. More generally, we prove the continuity of any Haar measurable polynomial map between locally compact groups, in the sense of Leibman. We deduce the above result as a direct consequence of a theorem about the automatic continuity of cocycles.