On the Effectiveness of Fekete’s Lemma in Information Theory

Abstract

Fekete’s lemma is a well known assertion that states the existence of limit values of superadditive sequences. In information theory, superadditivity of rate functions occurs in a variety of channel models, making Fekete’s lemma essential to the corresponding capacity problems. We analyze Fekete’s lemma with respect to effective convergence and computability and show that Fekete’s lemma exhibits no constructive derivation. In particular, we devise a superadditive, computable sequence of rational numbers so that the associated limit value in the sense of Fekete’s lemma is not a computable number. We further characterize the requirements for effective convergence and investigate the speed of convergence, as proposed by Rudolf Ahlswede in his 2006 Shannon lecture.