ARBITRARILY LONG ARITHMETIC PROGRESSIONS FOR CONTINUED FRACTIONS OF LAURENT SERIES

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 coefficients lying in F-q. In this paper, we concern with the analogous Szemeredi problem for continued fractions of Laurent series: we will show that the set of points x is an element of F-q((X-1)) of whose sequence of degrees of partial quotients is strictly increasing and contain arbitrarily long arithmetic progressions is of Hausdorff dimension 1/2.