On homogeneous and inhomogeneous Diophantine approximation over the fields of formal power series

Abstract

We prove a sharp analogue of Minkowski's inhomogeneous approximation theorem over fields of power series Fq((T −1 )). Furthermore, we study the approximation to a given point y in Fq((T −1 )) 2 by the SL 2 (Fq[T ])-orbit of a given point x in Fq((T −1 )) 2 .

On Hausdorff dimension in inhomogeneous Diophantine approximation over global function fields

“…The proof of Lemma 2.6 is similar to the one given after [BZ,Def. 3.3] in the particular case when K " F q pZq and v " 8, without weights.…”

“…The following corollary is due to [Kri,Theo. 1.1] (see also [BZ,Theo. 3.2] where the assumption that c m is divisible by n is implicit) in the special case when K " F q pZq and v " 8 and without weights.…”

“…In this subsection, we construct a subsequence with controlled growth of the best approximation sequence with weights of a matrix, assuming that its transpose is singular on average for those weights. We use as inspiration [BZ,page 470] in the special case of K " F q pZq and v " v 8 , and the first claim of the proof of [BuKLR,Theo. 2.2] in the disjoint case of the field Q.…”

“…The proofs of the above main theorems of this paper are largely divided into two parts. Firstly, assuming the singular on average property in order to prove the full Hausdorff dimension property, we give a lower bound on the Hausdorff dimension of appropriately chosen subsets of K m v , using new function fields versions of classical tools in Diophantine approximation such as geometry of numbers, transference principle and best approximation vectors (see for instance [Cas,Sch3,Kri,KlW,Cheu,Chev,GE,CC,KlST,GG,BZ,Ger,LSST,CGGMS,BuKLR]). Secondly, in order to prove the upper bound in Theorem 1.2, we use technics of homogeneous dynamics of diagonal actions and in particular the entropy method (see for instance [Kle, EL, LSS, ELW]).…”

The inhomogeneous Sprindžhuk conjecture over a local field of positive characteristic

“…In [22], the second named author proved the Sprindžhuk conjectures in this setting (in fact, also in multiplicative form), here we prove the inhomogeneous variant of the conjecture. We use the inhomogeneous transference principle of Beresnevich and Velani [8] to transfer the homogeneous result from [22] and also use a positive characteristic version of the transference principle of Bugeaud and Laurent interpolating between uniform and standard Diophantine exponents, established recently by Bugeaud and Zhang [10]. The possibility of proving the S-arithmetic inhomogeneous Sprindžhuk conjectures was suggested by Beresnevich and Velani ([8], §8.4) and the present paper realises this expectation in another natural setting, that of local fields of positive characteristic.…”