Counterexamples to the $p(t)$-adic Littlewood Conjecture Over Small Finite Fields

TL;DR

The paper studies the $p(t)$-adic Littlewood Conjecture ($p(t)$-$LC$) and its $t$-adic analogue over finite fields, proving an explicit counterexample to $t$-$LC$ in characteristic $5$ via the Second Level Paperfolding Sequence and conjecturing counterexamples in all characteristics $p\equiv1\pmod4$. It leverages the Number Wall framework, recasting Diophantine approximation into a combinatorial problem about Toeplitz-determinant arrays associated to coefficient sequences, and uses a computer-assisted approach that combines two-dimensional automatic sequences and morphisms. A key technical contribution is an efficient algorithm that constructs an explicit $[2,2]$-morphism and a tile-based coding to realize the number wall of the Paperfolding sequence, enabling verification through Frame Constraints. The results, supported by computational evidence for characteristics $7$ and $11$, suggest a unified pattern for the failure of $t$-$LC$ (and hence $p(t)$-$LC$) in odd characteristic and provide practical methods for exploring Littlewood-type conjectures over function fields.

Abstract

In 2004, de Mathan and Teulié stated the $p$-adic Littlewood Conjecture ($p$-$LC$) in analogy with the classical Littlewood Conjecture. Given a field $\mathbb{K}$ and an irreducible polynomial $p(t)$ with coefficients in $\mathbb{K}$, $p$-$LC$ admits a natural analogue over function fields, abbreviated to $p(t)$-$LC$ (and to $t$-$LC$ when $p(t)=t$). In this paper, an explicit counterexample to $p(t)$-$LC$ is found over fields of characteristic 5. Furthermore, it is conjectured that this Laurent series disproves $p(t)$-$LC$ over all fields of characteristic $p\equiv 1 \mod 4$. This fills a gap left by a breakthrough paper from Adiceam, Nesharim and Lunnon (2022) in which they conjecture $t$-$LC$ does not hold over all complementary fields of characteristic $p\equiv 3\mod 4$ and proving this in the case $p=3$. Supported by computational evidence, this provides a complete picture on how $p(t)$-$LC$ is expected to behave over all fields with characteristic not equal to 2. Furthermore, the counterexample to $t$-$LC$ over fields of characteristic 3 found by Adiceam, Nesharim and Lunnon is proven to also hold over fields of characteristic 7 and 11, which provides further evidence to the aforementioned conjecture. Following previous work in this area, these results are achieved by building upon combinatorial arguments and are computer assisted. A new feature of the present work is the development of an efficient algorithm (implemented in Python) that combines the theory of automatic sequences with Diophantine approximation over function fields. This algorithm is expected to be useful for further research around Littlewood-type conjectures over function fields.

Figures (11)

• Figure 1: Illustration of a window in a number wall. The window, inner frame and outer frame are in red, green and blue, respectively.
• Figure 2: The number wall of a sequence of length 25 generated uniformly and randomly over $\mathbb{F}_5$. The zero entries are coloured in red, with the nonzero values assigned a shade of grey, with 1 being the darkest and 4 being the lightest. The top row (red) has index $-2$. It is included as it initiates the recurrence relations given by Theorem \ref{['FC']} that are used to generate this image.
• Figure 3: The shape of a finite number wall generated by $\tau(\sigma)$ when $L(\tau(\sigma))=8$. Each block represents an entry, with the dark blue representing the rows with index $m\le -1$ and the longest row being the sequence given by $\tau(\sigma)$.
• Figure 4: A large potion of the number wall split into tiles depending on if $\tau(\sigma)$ has odd or even length. The dark red and blue show where the row index becomes negative. The remaining colours serve no other purpose than to distinguish one tile from another.
• Figure 5: Left: The number wall of a sequence split into tiles. Any tiles that are entirely contained on rows of negative index are coloured in black and grey. The lighter row in the middle denotes the zeroth row of the number wall. The remaining colours serve no other purpose than to distinguish one tile from another. Right: The two dimensional sequence $\mathcal{T}$ that is identified with the number wall.

Theorems & Definitions (31)

• Conjecture 1.1: $p$-$LC$, de Mathan and Teulié, 2004
• Conjecture 1.2: $p(t)$-$LC$
• Definition 1.3
• Conjecture 1.4: Adiceam, Nesharim, Lunnon
• Theorem 1.5
• Theorem 1.6
• Corollary 1.7
• Conjecture 1.8
• Theorem 1.9
• Definition 2.1