  A classical problem in number theory is showing that the mean value of an
arithmetic function is asymptotic to its mean value over a short interval or
over an arithmetic progression, with the interval as short as possible or the
modulus as large as possible.
  We study this problem in the function field setting, and prove for a wide
class of arithmetic functions (namely factorization functions), that such an
asymptotic result holds, allowing the size of the short interval to be as small
as a square-root of the size of the full interval, and analogously for
arithmetic progressions.
  For instance, our results apply for the indicator function of polynomials
with a divisor of given degree, and are much stronger than those known for the
analogous function over the integers.
  As opposed to many previous works, our results apply in the
\emph{large-degree limit}, where the base field $\mathbb{F}_q$ is fixed. Our
proofs are based on relationships between certain character sums and symmetric
functions, and in particular we use results from symmetric function theory due
to E\u{g}ecio\u{g}lu and Remmel. We also use recent bounds of Bhowmick, L\^{e}
and Liu on character sums, which are in the spirit of the
Drinfeld--Vl\u{a}du\c{t} bound.
