Remarks on Diophantine approximation in function fields

Abstract

We study some problems in metric Diophantine approximation over local fields of positive characteristic.

Singular vectors in affine subspaces and $Ψ$-Dirichlet numbers

“…Our Theorem 2.4 generalizes the above mentioned result and shows that even ψ-Dirichlet numbers are also only rational numbers. We note that the technique of [27] is different than ours. (4) When K = Q ν , there is no suitable continued fraction estimate that allows us to study the above theorem with the same techniques.…”

“…(3) Theorem 2.4 in [27] shows that only Dirichlet improvable numbers in function field are rational functions. Our Theorem 2.4 generalizes the above mentioned result and shows that even ψ-Dirichlet numbers are also only rational numbers.…”

“…Recent developments in ν-adic Diophantine approximation can be found in [42,53,6,21,5] and references therein. We refer the reader to the survey [24,48] and [1,24,28,27,34,4,47,56,26,25] for more recent results in Diophantine approximation in positive characteristic.…”

“…In fact, the converse is also true when n = 1 and K = R, F q ((T −1 )) (ref. [13], [27]). Definition 1.3.…”

The inhomogeneous Sprindžhuk conjecture over a local field of positive characteristic

“…This subject has been extensively studied, beginning with work of E. Artin [1] who developed the theory of continued fractions, and continuing with Mahler who developed Minkowski's geometry of numbers in function fields and Sprindžuk who, in addition to proving the analogue of Mahler's conjectures, also proved some transference principles in the function field setting (see [42]). The subject has also received considerable attention of late, we refer the reader to [15,37] for overviews and to [2,28,18,36,29] for a necessarily incomplete set of references.…”