On T-Adic Littlewood Conjecture For Certain Infinite Products

We consider a Laurent series defined by the infinite product g(u)(t) = Pi(infinity)(n=0)(1+ut(-2n)), where u is an element of F is a parameter and F is a field. We show that for all u is an element of Q \ {-1, 0, 1} the series g(u)(t) does not satisfy the t-adic Littlewood conjecture. On the other hand, if F is finite then g(u)(t) is an element of F((t(-1))) is either rational or it satisfies the t-adic Littlewood conjecture.