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

Establish that for every prime p, every irreducible polynomial P(t) in F_p[t], and every quadratic irrational Laurent series Θ(t) in F_p((t^{-1})), the sequence {P(t)^k}_{k≥0} has full maximal escape of mass for Θ(t). Concretely, prove that lim_{n→∞} limsup_{k→∞} (∑_{i=1}^{ℓ_{Θ·P^k}} max{deg(A_i^{[Θ·P^k]}(t)) − n, 0}) / (∑_{i=1}^{ℓ_{Θ·P^k}} deg(A_i^{[Θ·P^k]}(t))) = 1, where [A_0^{[Θ·P^k]}(t); A_1^{[Θ·P^k]}(t), …] denotes the continued fraction expansion of Θ(t)·P(t)^k and ℓ_{Θ·P^k} is the period length of its eventually periodic part.

Background

The classical KPS conjecture asserted full escape of mass for quadratic irrationals over finite fields along rational Hecke branches; this paper, together with recent work for p=2, provides counterexamples, disproving that conjecture. Motivated by these failures, the authors introduce a strengthened notion—maximal escape of mass—based on a limsup formulation of the escape proportion measured via degrees of partial quotients in continued fractions.

They prove full maximal escape of mass for specific automatic sequences (the p-Cantor sequence) and cite analogous results for Thue–Morse, which collectively motivate a general conjecture that all quadratic irrationals should exhibit full maximal escape of mass with respect to any irreducible polynomial sequence {P(t)^k}. This would restore a universal positive result in the refined setting.