On the theorem of M.Golomb

Abstract

Let $X_1,...,X_n$ be compact spaces and $X=X_1\times ...\times X_n.$ Consider the approximation of a function $f\in C(X)$ by sums $g_1(x_1)+...g_n(x_n),$ where $g_i\in C(X_i),$ $i=1,...,n.$ In [8], M.Golomb obtained a formula for the error of this approximation in terms of measures constructed on special points of $X$, called "projection cycles". However, his proof had a gap, which was pointed out by Marshall and O'Farrell [15]. But the question if the formula was correct, remained open. The purpose of the paper is to prove that Golomb's formula holds in a stronger form.

8 pages; the previous main result has substantially improved