New Bounds on Antipowers in Words

TL;DR

This paper investigates bounds on the minimal length of binary words that necessarily contain either a k-power or an r-antipower, providing new upper and lower bounds and extending to larger alphabets.

Contribution

It introduces new bounds on N(k, r), the minimal length ensuring the presence of k-powers or r-antipowers in binary words, and explores antipower avoidance over larger alphabets.

Findings

New upper and lower bounds on N(k, r) for binary words.

Lower bounds for N(k, 5) over larger alphabets.

Analysis of avoiding 3- and 4-antipowers in larger alphabets.

Abstract

Fici et al. defined a word to be a k-power if it is the concatenation of k consecutive identical blocks, and an r-antipower if it is the concatenation of r pairwise distinct blocks of the same size. They defined N (k, r) as the smallest l such that every binary word of length l contains either a k-power or an r-antipower. In this note we obtain some new upper and lower bounds on N (k, r). We also consider avoiding 3-antipowers and 4-antipowers over larger alphabets, and obtain a lower bound for N (k, 5) in the binary case.