Combinatorics on number walls and the P (t) P (t) ‐adic Littlewood conjecture

Abstract In 2004, de Mathan and Teulié stated the ‐adic Littlewood conjecture (‐LC) in analogy with the classical Littlewood conjecture. Let be a finite field be an irreducible polynomial with coefficients in . This paper deals with the analogue of ‐LC over the ring of formal Laurent series over , known as the ‐adic Littlewood conjecture (‐LC). First, it is shown that any counterexample to ‐LC for the case induces a counterexample to ‐LC when is any irreducible polynomial. Since Adiceam, Nesharim and Lunnon (2021) disproved ‐LC when and when is a finite field with characteristic 3, one obtains a disproof of ‐LC over any such field in full generality (i.e., for any choice of irreducible polynomial ). The remainder of the paper is dedicated to proving two metric results on ‐LC with an additional monotonic growth function over an arbitrary finite field. The first — a Khintchine‐type theorem for ‐adic multiplicative approximation — enables one to determine the measure of the set of counterexamples to ‐LC for any choice of . The second complements this by showing that the Hausdorff dimension of the same set is maximal in the critical case where . These results are in agreement with the corresponding theory of multiplicative Diophantine approximation over the reals. Beyond the originality of the results, the main novelty of the work comes from the methodology used. Classically, Diophantine approximation employs methods from either Number Theory or Ergodic Theory. This paper provides a third option: combinatorics. Specifically, an extensive combinatorial theory is developed relating ‐LC to the properties of the so‐called number wall of a sequence. This is an infinite array containing the determinant of every finite Toeplitz matrix generated by that sequence. In full generality, the paper creates a dictionary allowing one to transfer statements in Diophantine approximation in positive characteristic to combinatorics through the concept of a number wall, and conversely.