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

In 2004, de Mathan and Teuli\'e 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.