Almost orthogonally additive functions

Abstract

If a function $f$, acting on a Euclidean space ${\mathbb{R}}^n$, is "almost" orthogonally additive in the sense that $f(x+y)=f(x)+f(y)$ for all $(x,y)\in \perp \setminus Z$, where $Z$ is a "negligible" subset of the $(2n-1)$-dimensional manifold $\perp \subset {\mathbb{R}}^{2n}$, then $f$ coincides almost everywhere with some orthogonally additive mapping.