On Effective Convergence in Fekete's Lemma and Related Combinatorial Problems in Information Theory
Holger Boche, Yannik Böck, Christian Deppe
TL;DR
The paper investigates the computability of limits arising from Fekete's lemma by embedding them in the Zheng–Weihrauch arithmetical hierarchy of real numbers. It develops monotone representations and shows that $Σ_1$ and $Π_1$ precisely characterize limits produced by superadditive and subadditive sequences, respectively, implying general noncomputability of Fekete limits. For nonnegative, computable sequences, it provides an effective equivalence: the limit is computable iff the convergence is effective, and it offers a constructive modulus of convergence. It also demonstrates a separation between computable sequences of rationals and computable sequences of computable numbers, and concludes that Fekete's lemma alone cannot yield computable channel capacities in general, with implications for information-theoretic multi-letter formulas and their computability.
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 \(Σ_1\) and \(Π_1\). 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.