ARBITRARILY LONG ARITHMETIC PROGRESSIONS FOR CONTINUED FRACTIONS OF LAURENT SERIES

摘要

A famous theorem of Szemer'edi asserts that any subset of integers with positive upper density contains arbitrarily arithmetic progressions.Let F q be a finite field with q elements and F q((X 1)) be the power field of formal series with coefcients lying in F q.In this paper,we concern with the analogous Szemer′edi problem for continued fractions of Laurent series: we will show that the set of points x ∈ F q((X 1)) of whose sequence of degrees of partial quotients is strictly increasing and contain arbitrarily long arithmetic progressions is of Hausdorf dimension 1/2.