On t-adic Littlewood conjecture for generalised Thue-Morse functions

Abstract

We consider a Laurent series defined by infinite products $g_u(t)={\prod }_{n=0}^{\infty }(1+u{t}^{-2^n})$, 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 a rational function or it satisfies the $t$-adic Littlewood conjecture.