Combinatorics on Number Walls and the $P(t)$-adic Littlewood Conjecture

For any prime $p$ and real number and $\alpha$, the $p$-adic Littlewood Conjecture due to de Mathan and Teuli\'e asserts that $$\inf_{|m|\ge1}|m|_p\cdot |m|\cdot |\left\langle\alpha m\right\rangle|=0.$$ Above, $|m|$ is the usual absolute value, $|m|_p$ is the $p$-adic norm and $\left|\left\langle x\right\rangle\right|$ is the distance from $x\in\mathbb{R}$ to the nearest integer. Let $\mathbb{K}$ be a field and $P(t)\in\mathbb{K}[t]$ be an irreducible polynomial. This paper deals with the analogue of this conjecture over the field of formal Laurent series over $\mathbb{K}$, 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 $\mathbb{K}$ 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 $\log^2$. 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.