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

2020/01/06 by Dzmitry Badziahin, Badziahin, Dzmitry
Mathematics · #Mathematical Dynamics and Fractals #Algebraic Geometry and Number Theory #Advanced Mathematical Identities

paper · pdf · doi:10.48550/arxiv.2001.01422

Abstract

We consider a Laurent series defined by infinite products g_u(t) = ∏_n=0^∞ (1 + ut^-2n), where u∈ \mathbbF is a parameter and \mathbbF is a field. We show that for all u∈ℚ∖\-1,0,1\ the series g_u(t) does not satisfy the t-adic Littlewood conjecture. On the other hand, if \mathbbF is finite then g_u(t)∈ \mathbbF((t^-1)) is either a rational function or it satisfies the t-adic Littlewood conjecture.

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