Equation $x^iy^jx^k=u^iv^ju^k$ in words

TL;DR

This paper proves conditions under which certain words in formal language theory are periodicity forcing and demonstrates the optimality of these bounds with examples.

Contribution

It establishes new criteria for when specific words enforce periodicity, including optimal bounds, in formal language theory.

Findings

Words of the form a^i b^j a^k are periodicity forcing if j ≥ 3 and i + k ≥ 3.

Both bounds for j and i + k are proven to be optimal.

Provides examples confirming the bounds are tight.

Abstract

We will prove that the word $a^ib^ja^k$ is periodicity forcing if $j \geq 3$ and $i+k \geq 3$, where $i$ and $k$ are positive integers. Also we will give examples showing that both bounds are optimal.