A simpler proof of Sternfeld's Theorem

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.