Counterexamples to the 𝑝(𝑡)-adic Littlewood conjecture over small finite fields
Samuel Garrett, Steven Robertson · Mathematics of Computation · 2025
De Mathan and Teulié [Monatsh. Math. 143 (2004), pp. 229–245] stated the p p - adic Littlewood Conjecture ( p p - L C LC ) in analogy with the classical Littlewood Conjecture. Given a field K \mathbb {K} and an irreducible polynomial p ( t ) p(t) with coefficients in K \mathbb {K} , p p - L C LC admits a natural analogue over function fields, abbreviated to p ( t ) p(t) - L C LC (and to t t - L C LC when p ( t ) = t p(t)=t ). In this paper, an explicit counterexample to p ( t ) p(t) - L C LC is found over fields of characteristic 5. Furthermore, an infinite family of Laurent series is described that is conjectured to disprove