On Effective Convergence in Fekete's Lemma and Related Combinatorial Problems in Information Theory

TL;DR

This paper examines the computability of limits in Fekete's lemma within an information-theoretic framework, revealing that these limits are generally non-computable but can be effectively computed under certain conditions.

Contribution

It introduces a novel analysis of Fekete's lemma using the arithmetical hierarchy, characterizes when limits are computable, and explores implications for information theory.

Findings

Limits in Fekete's lemma are generally non-computable.

Effective computability of limits depends on sequence properties.

Structural differences between computable sequences of numbers and rationals are established.

Abstract

Fekete's lemma is a well known result from combinatorial mathematics that shows the existence of a limit value related to super- and subadditive sequences of real numbers. In this paper, we analyze Fekete's lemma in view of the arithmetical hierarchy of real numbers by Zheng and Weihrauch and fit the results into an information-theoretic context. We introduce special sets associated to super- and subadditive sequences and prove their effective equivalence to \(\Sigma_1\) and \(\Pi_1\). Using methods from the theory established by Zheng and Weihrauch, we then show that the limit value emerging from Fekete's lemma is, in general, not a computable number. Given a sequence that additionally satisfies non-negativity, we characterize under which conditions the associated limit value can be computed effectively and investigate the corresponding modulus of convergence. Subsidiarily, we prove a theorem concerning the structural differences between computable sequences of computable numbers and computable sequences of rational numbers. We close the paper by a discussion on how our findings affect common problems from information theory.