On sets of exact Diophantine approximation over the field of formal series

Abstract

Given a function ψ : R > 0 → R > 0 with ψ ( x ) = o ( x − 2 ) , let Exact ( ψ ) be the set of exact Diophantine approximation, namely the set of real numbers that are approximable by rational numbers to order ψ, but to no order cψ with 0 < c < 1 . It is unknown whether the set Exact ( ψ ) is empty, but when ψ is non-increasing the Hausdorff dimension of the set Exact ( ψ ) is known to be 2 / λ , where λ is the lower order at infinity of the function 1 / ψ . In this paper, over the field of Laurent series we prove the set of exact Diophantine approximation is uncountable, and when the error function is non-increasing we give its Hausdorff dimension analogous to the real case. Furthermore, we give a metric result about the set of Laurent series that are approximable to exact order.