Full generic escape of mass for all quadratic irrationals over finite fields

Establish that for every finite field F_q, every irreducible polynomial P(t) in F_q[t], and every quadratic irrational Laurent series Θ(t) in F_q((t^{-1})), Θ(t) exhibits full generic escape of mass with respect to the sequence {P(t)^k}_{k≥0}. Specifically, prove that for every ε > 0, lim_{n→∞} d({k ∈ ℕ : e_{k,n}(Θ(t)) > 1 − ε}) = 1, where e_{k,n}(Θ(t)) denotes the proportion of mass above threshold n computed from the degrees of the periodic partial quotients of the continued fraction of Θ(t)·P(t)^k, and d(·) is natural density.

Background

Beyond pointwise liminf or limsup considerations, the authors introduce a metric notion—generic escape of mass—measured by natural density of indices k whose escape proportion is arbitrarily close to 1. They prove full generic escape of mass for two benchmark automatic sequences (the p-Cantor and Thue–Morse sequences), suggesting this behavior may be widespread.

Leveraging these results and prior work on Hecke dynamics and escape of mass, the authors formulate a conjecture asserting that every quadratic irrational over F_q exhibits full generic escape of mass along the sequence {P(t)^k}, thus proposing a strong metric analogue of the original (now false) KPS-type statements.