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Escape of Mass of the $p$-Cantor Sequence

Noy Soffer Aranov, Steven Robertson

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.

Escape of Mass of the $p$-Cantor Sequence

Abstract

Let be a prime. In 2017, Kemarsky, Paulin, and Shapira (KPS) conjectured that any Laurent series over exhibits full escape of mass with respect to any irreducible polynomial . In 2025, this was shown to be false in the case and by Nesharim, Shapira and the first named author. This work shows that for any odd prime and any irreducible polynomial , the so-called -Cantor sequence provides a counterexample to the aforementioned conjecture over . 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.
Paper Structure (57 sections, 43 theorems, 172 equations, 5 figures)

This paper contains 57 sections, 43 theorems, 172 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Continued Fraction Expansions in $\mathbb{F}_q(\!(t^{-1})\!)$
  3. Escape of Mass in Hecke Trees over $\mathbb{F}_q(\!(t^{-1})\!)$
  4. Main Results
  5. Disproving Conjecture \ref{['conj:KPS']}
  6. Maximal Escape of Mass
  7. Generic Escape of Mass
  8. Structure of Paper
  9. Establishing Theorem \ref{['thm: t_to_P(t)']}
  10. Number Walls
  11. Definition of a Number Wall
  12. The Profile of a Number Wall
  13. Diagonals of Number Walls
  14. Identifying Sequences with Laurent Series
  15. Escape of Mass in Number Walls
  16. ...and 42 more sections

Key Result

Theorem 1.1

AS Let $p$ be a prime and let $\alpha$ be a quadratic irrational. Then, there exists a constant $c$ depending on $\alpha$ and on $p$, such that for every $k\in \mathbb{N}$,

Figures (5)

  • Figure 1: Left: The number wall $W_5(\textbf{S})$ for the sequence $\textbf{S}=(1,1,3,2,1,0,0,0,2,0,2,0)\in\mathbb{F}_5^{12}$, where the zeroes are coloured in red. Right: The profile of $W_\mathbb{K}(\textbf{S})$.
  • Figure 2: The profile of the first three diagonals of a number wall are depicted above, with the top row having index 0. The first, second and third diagonal are drawn in black, blue and red respectively. The degree of $A_i^{[\Theta\cdot t^k]}(t)$ is given by one plus the number of zeroes between the $i^\textup{th}$ and the $(i+1)^\textup{th}$ nonzero entry of diagonal $k$.
  • Figure 3: An illustration of diagonal $k$ of the number wall. The large dots represent a generic entry of $\mathbb{K}$. The entry on row $-1$ is nonzero from Definition \ref{['nw']}.
  • Figure 4: The $j$-th diagonal is periodic, with its period being comprised of its journey through four components, where the red part is its intersection with $\Phi^k(A)$, the cyan part is its intersection with $Z_k$, the pink part is its intersection with $\Phi^k(\iota(A))$, and the blue part is its intersection with $W_k$.
  • Figure 5: The path of the $j$-th diagonal in $\Phi_p^{k+1}(A)$. The purple diagonal corresponds to even $j_k$, whereas the pink diagonal corresponds to odd $j_k$.

Theorems & Definitions (110)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Conjecture 1.8
  • Theorem 1.9
  • Definition 1.10
  • ...and 100 more