An application of Kronecker's theorem to rational functions (Q1318127)

Let \(\sum^ \infty_{n=0} a_ n z^ n\) be the Taylor series of a rational function regular at zero. Suppose that the limit set \(L(z)\) of the sequence of the partial sums \(\sum^ N_{n=0} a_ n z^ n\), \(N=0,1,\dots,\) is contained in a finite union of circles, for any \(z\) belonging in a ``large'' subset \(E\) of the unit circle. Using Kronecker's theorem in diophantine approximation it is proven here that \(\sum^ \infty_{n=0} a_ n z^ n\) can be decomposed as a sum of two power series; one of them converges on \(E\) and the partial sums of the other lie on a finite union of circles.