On the Littlewood conjecture in fields of power series

Abstract

Let k be an arbitrary field. For any fixed badly approximable power series Θ in k((X −1 )), we give an explicit construction of continuum many badly approximable power series Φ for which the pair (Θ, Φ) satisfies the Littlewood conjecture. We further discuss the Littlewood conjecture for pairs of algebraic power series.

Dirichlet's Theorem in Function Fields

“…For instance the theory of badly approximable numbers and vectors in positive characteristic offers several surprises: there is no analogue of Roth's theorem, provided that the base field is finite, which we assume throughout this paper. We refer the reader to [2] for other results in this vein.…”

An unresolved analogue of the Littlewood Conjecture

“…For many years, mathematicians accepted this claim. Armitage's proof appeared to imitate Baker's proof for his counterexample to the analogue of the Littlewood Conjecture with K = ‫.ޒ‬ However, we found a parenthetical comment in [Adamczewski and Bugeaud 2007] that Armitage's counterexample does not hold, an observation these authors attribute to Bernard de Mathan. We also found a reference in [Larcher and Niederreiter 1993] that Yves Taussat, a student of Mathan, disproved Armitage's claim in his Ph.D. thesis [Taussat 1986].…”

Multiplicatively badly approximable matrices in fields of power series

“…Let K be a field and let L = K((x −1 )) be the field of a formal Laurent series with coefficients in K. That is, the nonzero elements of L consist of all formal Laurent series (1) F…”

“…where F (x) = 0 L in L is given by (1). It follows that | | : L → [0, ∞) is a discrete, non-archimedean absolute value on L, and the resulting metric space L, | | is complete.…”

“…We can formulate Diophantine approximation problems in this setting, with K[x], K(x), L, and T, playing roles analogous to Z, Q, R, and the quotient group R/Z, respectively. In the case where K is finite, the analog of the Littlewood conjecture in L is believed to be true, but still remains an open problem (see [1]). Explicit counterexamples were later given by Baker [2].…”