Decomposition of number arrangements in the cube

TL;DR

This paper characterizes basic subsets in three-dimensional space where functions can be decomposed into sums of single-variable functions, providing criteria and constructions for minimal non-basic subsets based on graph properties.

Contribution

It introduces a criterion for basic subsets in specific cases and constructs minimal non-basic subsets using graph-theoretic methods.

Findings

Established a criterion for basic subsets in certain cases

Developed constructions for minimal non-basic subsets

Connected subset properties to graph theory

Abstract

A subset $M \subset \textbf{R}^3$ is called a \emph{basic subset}, if for any funciton $f \colon M \to \textbf{R}$ there exist such functions $f_1; f_2; f_3 \colon \textbf{R} \to \textbf{R}$ that $f(x_1, x_2, x_3) = f_1(x_1) + f_2(x_2) + f_3(x_3)$ for each point $(x_1, x_2, x_3)\in M$. In this article we prove a criterion for a basic subset for some specific subsets in terms of some graph properties. We also introduce several constructions for mimimal non-basic subsets. The article is written in Russian.