On $t$-adic Littlewood conjecture for certain infinite products

Abstract:

We consider a Laurent series defined by the infinite product $g_u(t) = \prod _{n=0}^\infty (1 + ut^{-2^n})$, where $u\in \mathbb {F}$ is a parameter and $\mathbb {F}$ is a field. We show that for all $u\in \mathbb {Q}\setminus \{-1,0,1\}$ the series $g_u(t)$ does not satisfy the $t$-adic Littlewood conjecture. On the other hand, if $\mathbb {F}$ is finite then $g_u(t)\in \mathbb {F}((t^{-1}))$ is either rational or it satisfies the $t$-adic Littlewood conjecture.