Dirichlet's Theorem in Function Fields

Abstract

Abstract. We study metric Diophantine approximation for function elds speci cally the problem of improving Dirichlet's theorem in Diophantine approximation.

Remarks on Diophantine approximation in function fields

“…A point x ∈ F d is said to be (k, ε) − Dirichlet improvable if there exists m 0 ∈ N such that for every m ≥ m 0 , the following system of inequations admits a nonzero solution P ∈ Λ[X 1 , ..., X d ] with total degree ≤ k |P (x)| < ε e mN H(P ) < εe m . When k = 1, this coincides with the usual Dirichlet improvable vectors in F d , studied in [9]. Furthermore, one has the following observation in the case d = 1.…”

“…We first prove the following analogue of the Dirichlet's theorem in this set up. The above theorem indeed almost follows from Theorem 2.1 of [9]. In fact, it is just a restatement of [9, Theorem 2.1] if we consider H(P ).…”

“…where f is defined in (2.6), is a Dirichlet ε-improvable vector in F N in the sense of [9]. To see this, let us write the N monomials in the variables X 1 , · · · , X d having total degree ≤ k as M 1 (X 1 , · · · , X d ), · · · , M N (X 1 , · · · , X d ) so that a polynomial P ∈ Λ[X 1 , · · · , X d ] will look like…”

“…In it, the authors consider certain interesting Diophantine problems arising from K. R. Yu's [25] generalisation to higher dimensions, of Mahler's [21] classification of complex numbers according to their Diophantine properties. In this paper, which can be thought of as a sequel to our earlier work [9] and the paper [10] of the second named author, we consider function field analogues of their results. We recall Mahler's classification briefly for the reader, see [2] for details.…”

“…The subsequent sections are devoted, respectively, to the notion of k Dirichlet improvable vectors and k very well approximable vectors. Dirichlet improvable vectors and very well approximable vectors have been studied in the function field setting in a previous paper of the authors [9] and in a paper [10] of the second named author respectively. Following the ideas in [15], we explain how one can use these previously established results to study the refined notion of k-Dirichlet improvable and k-very well approximable vectors.…”