Escape of Mass of the $p$-Cantor Sequence

TL;DR

This paper demonstrates that for any odd prime, the $p$-Cantor sequence counterexamples disprove a conjecture about full escape of mass in Laurent series over finite fields, and introduces new concepts of escape of mass.

Contribution

It shows that the $p$-Cantor sequence provides counterexamples to the KPS conjecture for all odd primes and introduces the concepts of maximal and generic escape of mass.

Findings

Counterexamples for all odd primes $p$.

Introduction of maximal and generic escape of mass.

Validation of modified conjectures for previous counterexamples.

Abstract

Let $p$ be a prime. In 2017, Kemarsky, Paulin, and Shapira (KPS) conjectured that any Laurent series over $\mathbb{F}_p$ exhibits full escape of mass with respect to any irreducible polynomial $P(t)\in\mathbb{F}_p[t]$. In 2025, this was shown to be false in the case $p=2$ and $P(t)=t$ by Nesharim, Shapira and the first named author. This work shows that for any odd prime $p$ and any irreducible polynomial $P(t)\in\mathbb{F}_p[t]$, the so-called $p$-Cantor sequence provides a counterexample to the aforementioned conjecture over $\mathbb{F}_p$. Furthermore, the concepts of maximal escape of mass and generic escape of mass are introduced. These lead to two natural variations of the KPS conjecture, both of which are shown to hold for all previous counterexamples.