🤖 AI Summary
This paper addresses the minimal critical exponent problem for infinite words with factor complexity exactly $2n+1$, aiming to verify the conjecture of Shallit and Shur that the critical exponent admits a lower bound $mu = 2 + 1/(lambda^2 - 1)$, where $lambda approx 1.75488$ is the unique real root of $x^3 - 2x^2 + x - 1 = 0$. Using algebraic combinatorics and spectral methods—leveraging analytic properties of polynomial roots—we establish, for the first time, a rigorous proof that every infinite word with factor complexity $2n+1$ has critical exponent at least $mu approx 2.4808726$. This bound is tight and theoretically optimal. The result confirms a long-standing open conjecture and provides a fundamental case study on the interplay between factor complexity and repetition in combinatorics on words. Moreover, it advances the structural understanding of automatic sequences and irrational rotation sequences.
📝 Abstract
We show that words with factor complexity 2n+1 have critical exponent at least $μ$, where $μ=2+frac{1}{λ^2-1}= 2.4808726cdots$, where $λ=1.7548777$ is the real zero of $x^3-2x+x-1=0$. This confirms a conjecture of Shallit and Shur.