On the number of Dejean words over alphabets of 5, 6, 7, 8, 9 and 10 letters

TL;DR

This paper provides precise estimates of the growth rates of Dejean words over alphabets of sizes 5 to 10, demonstrating their exponential growth by combining new lower bounds with existing upper bounds.

Contribution

It introduces new lower bounds on the growth rates of Dejean words for alphabets of sizes 5 to 10, refining the understanding of their exponential growth.

Findings

Estimates of growth rates with 0.005 precision

Confirmation of exponential growth of Dejean words for k=5 to 10

Combined bounds narrow the growth rate range

Abstract

We give lower bounds on the growth rate of Dejean words, i.e. minimally repetitive words, over a k-letter alphabet, for k=5, 6, 7, 8, 9, 10. Put together with the known upper bounds, we estimate these growth rates with the precision of 0,005. As an consequence, we establish the exponential growth of the number of Dejean words over a k-letter alphabet, for k=5, 6, 7, 8, 9, 10.