A simpler proof of Sternfeld's Theorem

TL;DR

This paper offers a simplified proof of Sternfeld's Theorem, which characterizes when certain subsets of Euclidean space can be represented through sums of continuous functions, based on the presence of Sternfeld arrays.

Contribution

It introduces a more straightforward proof of Sternfeld's Arrays Theorem, enhancing understanding of basic sets in the context of Kolmogorov's Superposition Theorem.

Findings

A subset containing arbitrarily large Sternfeld arrays is not basic.

The paper presents a simpler proof of Sternfeld's Theorem.

Clarifies the role of Sternfeld arrays in the structure of basic sets.

Abstract

In Sternfeld's work on Kolmogorov's Superposition Theorem appeared the combinatorial-geometric notion of a basic set and a certain kind of arrays. A subset $X \subset \mathbb R^n$ is basic if any continuous function $X\to \mathbb R$ could be represented as the sum of compositions of continuous functions $\mathbb R\to \mathbb R$ and projections to the coordinate axes. The definition of a Sternfeld array will be presented in the paper. Sternfeld's Arrays Theorem. If a closed bounded subset $X \subset \mathbb R^{2n}$ contains Sternfeld arrays of arbitrary large size then $X$ is not basic. The paper provides a simpler proof of this theorem.