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KPS escape-of-mass conjecture in positive characteristic

Prove that for every irreducible polynomial P(t) in the polynomial ring F_q[t] over a finite field F_q and every quadratic irrational Laurent series Θ(t) in F_q((t^{-1})), the limit lim_{n→∞} liminf_{k→∞} ( max_{1≤i≤ℓ_k} { deg(A_i^{[Θ·P^k]}(t)) − n, 0 } ) / ( Σ_{i=1}^{ℓ_k} deg(A_i^{[Θ·P^k]}(t)) ) equals 1, where ℓ_k denotes the length of the periodic part of the continued fraction expansion of P^k(t)·Θ(t).

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Background

Over the reals, quadratic irrationals have eventually periodic continued fractions, and no single term captures a positive proportion of ‘mass’ along prime powers; however, in function fields Paulin–Shapira proved a strong ‘escape of mass’ phenomenon. Motivated by this, Kemarsky–Paulin–Shapira formulated a conjecture asserting maximal escape of mass (value 1 in the limiting ratio) for all irreducible P(t) and all quadratic irrational Laurent series Θ(t).

This paper is the first of two aimed at disproving that conjecture, but it quotes the conjecture explicitly as originally posed in the literature, providing the formal target and context for the results developed here.

References

Conjecture [Conjecture 6]{KPS} For every irreducible polynomial P(t)\in \mathbb{F}q[t] and every quadratic irrational Laurent series \Theta(t)\in \mathbb{F}_q(!(t{-1})!), one has \lim{n\rightarrow \infty}\liminf_{k\rightarrow \infty}\frac{\max\left{deg\left(A_{i}{[\Theta\cdot Pk]}(t)\right)-n,0\right}}{\sum_{i=1}{\ell_{k}}deg\left(A_{i}{[\Theta\cdot Pk]}(t)\right)}=1, where, \ell_k is the length of the periodic part of the continued fraction expansion of Pk(t)\cdot \Theta(t).

— Fractals Emerging from the Toepltiz Determinants of the p-Cantor Sequence (2510.19449 - Robertson et al., 22 Oct 2025) in Conjecture (KPS Conjecture 6), Introduction, Subsection “Escape of Mass over F_q((t^{-1}))” (Section 1.1.2)