Algebra and Number Theory Seminar: Combinatorics on Number Walls and the p(t)-adic Littlewood Conjecture

Abstract: The p-adic Littlewood Conjecture is one of the most well-studied open problems in Diophantine approximation. In this talk, we will discuss its analogue over fields of positive characteristic, known as the p(t)-adic Littlewood conjecture (p(t)-LC). In 2021, Adiceam, Nesharim and Lunnon disproved p(t)-LC in the special case that p(t)=t and the field has characteristic 3. We will show that this induces a counterexample to p(t)-LC for any polynomial p(t). Furthermore, a Khintchine-type theorem is acquired that yields the measure of the set of counterexamples to p(t)-LC when an additional growth function is added. The Hausdorff dimension of the same set is calculated when the growth function is specialised to log^2. These results are acquired by rephrasing p(t)-LC in terms of combinatoric statements on the so-called Number Wall of a sequence. This is a relatively unknown, yet extremely visual area of mathematics that has applications far beyond Diophantine approximation. This talk aims to be accessible to viewers who have never studied Diophantine approximation or number walls.