Almost orthogonally additive functions

TL;DR

This paper investigates functions on Euclidean spaces that are nearly orthogonally additive, showing they coincide almost everywhere with truly orthogonally additive functions despite being only approximately additive outside a negligible set.

Contribution

It establishes that functions approximately additive on orthogonal pairs are essentially orthogonally additive almost everywhere, extending the understanding of additive functions under near-conditions.

Findings

Functions almost orthogonally additive coincide with orthogonally additive functions almost everywhere.

Negligible sets where additivity fails do not affect the overall structure of the function.

The result applies to functions defined on Euclidean spaces with negligible exceptions.

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 $(x,y)\in\bot\setminus Z$, where $Z$ is a "negligible" subset of the $(2n-1)$-dimensional manifold $\bot\subset\mathbb{R}^{2n}$, then $f$ coincides almost everywhere with some orthogonally additive mapping.