Complexity and recurrence in infinite words and related structures

Abstract

We study the asymptotics and fine-scale behavior of quantitative combinatorial measures of infinite words and related dynamical and algebraic structures. We construct infinite recurrent words $w$ whose complexity functions $p_w(n)$ are arbitrarily close to linear, but whose discrete derivatives are not bounded from above by $p_w(n)/n$. Moreover, we construct words of polynomially bounded complexity whose discrete derivatives exceed $p_w(n)/n^\varepsilon$ infinitely often, for every given $\varepsilon>0$. These provide negative answers in a strong sense to an open question of Cassaigne from 1997, showing that his theorem on words of linear complexity is best possible. Next, we characterize, up to a linear multiplicative error, the complexity functions of strictly ergodic subshifts, showing that every non-decreasing, submultiplicative function arises in this setting. This gives the first `industrial' construction of strictly ergodic subshifts of prescribed subexponential complexity. We then investigate quantitative recurrence in uniformly recurrent words and, as an application, address a question of Bavula from 2006 related to holonomic inequalities on the spectrum of possible filter dimensions of simple associative algebras: we construct simple algebras of prescribed filter dimension in $[1,\infty)$ and essentially settling the problem entirely in the graded case. Throughout, we construct uniformly recurrent words of linear complexity and with arbitrary polynomial recurrence growth.