  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 Adiceam,
Nesharim and Lunnon proved the $p(t)$-LC is false when $p(t)=t$ and
$\mathbb{K}$ has characteristic three, 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 $p(t)$-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.
  (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.
