Full generic escape-of-mass conjecture for quadratic Laurent series

Let $q$ be a prime power, let $\Theta(t)\in\mathbb{F}_q(\!(t^{-1})\!)$ be quadratic irrational, and let $P(t)\in\mathbb{F}_q[t]$ be irreducible. For $n,k\in\mathbb{N}$, define

$$e_{k,n}(\Theta(t)):=\frac{\sum_{i=1}^{\ell_{\Theta\cdot P^k}}\max\left\{\operatorname{deg}\left(A_i^{[\Theta\cdot P^k]}(t)\right)-n,0\right\}}{\sum_{i=1}^{\ell_{\Theta\cdot P^k}}\operatorname{deg}\left(A_i^{[\Theta\cdot P^k]}(t)\right)}.$$

Full generic escape of mass means that for every $\varepsilon>0$,

$$\lim_{n\rightarrow\infty}d\left(\left\{k\in\mathbb{N}:e_{k,n}(\Theta(t))>1-\varepsilon\right\}\right)=1,$$

where $d$ denotes the natural density. Full generic escape-of-mass conjecture. The series $\Theta(t)$ exhibits full generic escape of mass with respect to the sequence $\{P(t)^k\}_{k\geq 0}$.

The paper proves the corresponding assertion for the $p$-Cantor sequence with $P(t)=t$ in odd characteristic, and cites the analogous Thue–Morse result. The general polynomial and quadratic-irrational statement remains open in the supplied text.