Revisiting Fekete's Lemma, Subadditive and Periodic Sequences

TL;DR

This paper offers a new proof of Fekete's Lemma, constructs subadditive functions interpolating sequences, and explores properties of periodic and approximately periodic sequences with stability and characterization results.

Contribution

It provides an alternative proof of Fekete's Lemma, explicit formulas for subadditive minorants, and new characterizations of periodic and approximately periodic sequences.

Findings

Constructed subadditive functions interpolating sequences.

Derived explicit formulas for subadditive minorants.

Characterized periodic and approximately periodic sequences.

Abstract

In this paper, we present an alternative proof of Fekete's Lemma. We demonstrate that for any subadditive sequence, it is possible to construct a subadditive function that exactly interpolates the sequence. Using this result, along with Hille's theorem on subadditive functions, we naturally arrive at Fekete's Lemma. Additionally, we provide an explicit formula for determining the largest subadditive minorant of a given sequence. We explore a sandwich-type result and derive a discrete version of the Hyers-Ulam type stability theorem. For approximately periodic sequences, we offer a decomposition result. In the final section, we propose two characterization theorems for ordinary periodic sequences.