2023/07/03 by S. H. Robertson, Robertson, Steven · 1 citation
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #advanced mathematical theories
paper · doi:10.48550/arxiv.2307.00955
openalex publication_date 2023/07/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For any prime p and real number and α, the p-adic Littlewood Conjecture due to de Mathan and Teulié asserts that inf|m|≥1|m|p⋅ |m|⋅ |⟨αm⟩|=0. Above, |m| is the usual absolute value, |m|p is the p-adic norm and |⟨ x⟩| is the distance from x∈ℝ to the nearest integer. Let \mathbbK be a field and P(t)∈\mathbbK[t] be an irreducible polynomial. This paper deals with the analogue of this conjecture over the field of formal Laurent series over \mathbbK, known as the P(t)-adic Littlewood Conjecture (P(t)-LC). The following results are established: (1) Any counterexample to P(t)-LC for the case P(t)=t generates a counterexample when P(t) is any irreducible polynomial. Since P(t)-LC is knwon to be false when P(t)=t and \mathbbK has characteristic 0,3,5,7 and 11, one obtains a disproof of the P(t)-LC over any such field for any choice of irreducible polynomial P(t). (2) A Khintchine-type theorem for t-adic multiplicative approximation is established, enabling one to determine the measure of the set of counterexamples to P(t)-LC with an additional monotonic growth function in the case P(t)=t. (3) The Hausdorff dimension of the same set is shown to be maximal when P(t)=t in the critical case where the growth function is log2. These goals are achieved by developing an extensive theory in combinatorics relating P(t)-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. The main novelty of this paper is creating a dictionary allowing one to transfer statements in Diophantine approximation in positive characteristic to combinatorics through the concept of a number wall, and conversely.