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 q be a finite field with q elements and q((X−1)) be the power field of formal series with coefficients lying in q. In this paper, we concern with the analogous Szemerédi problem for continued fractions of Laurent series: we will show that the set of points x ∈ 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.