On sums of degrees of the partial quotients in continued fraction expansions of Laurent series

For any formal Laurent series x = ∑ n = v ∞ c n z − n with coefficients c n lying in some given finite field, let x = [ a 0 ( x ) ; a 1 ( x ) , a 2 ( x ) , … ] be its continued fraction expansion. It is known that, with respect to the Haar measure, almost surely, the sum of degrees of partial quotients deg a 1 ( x ) + ⋯ + deg a n ( x ) grows linearly. In this note, we quantify the exceptional sets of points with faster growth orders than linear ones by their Hausdorff dimension, which covers an earlier result by J. Wu.