The complexity bound conjecture for weakly mixing subshifts

Let $p(q)$ denote the word complexity of a subshift, namely the number of distinct words of length $q$, and suppose the subshift admits a weakly mixing probability measure. The quantity $\limsup \nicefrac{p(q)}{q}$ measures its asymptotic upper complexity ratio. Complexity bound conjecture. Every subshift admitting a weakly mixing probability measure has complexity such that

$$\limsup \nicefrac{p(q)}{q} > 1.5.$$

The conjecture asserts that the examples constructed in the paper, whose upper complexity ratio approaches $1.5$ from above, are optimal. In particular, no subshift admitting a weakly mixing probability measure can have upper complexity ratio strictly below $1.5$; whether the bound is sharp at exactly $1.5$ remains part of the surrounding open problem.