We develop an approach to study character sums, weighted by a multiplicative
function f:Fq[t]→S1, of the form \begin{equation} \sum_{G\in
\mathcal{M}_N}f(G)\chi(G)\xi(G), \end{equation} where χ is a Dirichlet
character and ξ is a short interval character over Fq[t]. We
then deduce versions of the Matom\"aki-Radziwill theorem and Tao's two-point
logarithmic Elliott conjecture over function fields Fq[t], where
q is fixed. The former of these improves on work of Gorodetsky, and the
latter extends the work of Sawin-Shusterman on correlations of the M\"{o}bius
function for various values of q.
Compared with the integer setting, we encounter a different phenomenon,
specifically a low characteristic issue in the case that q is a power of 2.
As an application of our results, we give a short proof of the function field
version of a conjecture of K\'atai on classifying multiplicative functions with
small increments, with the classification obtained and the proof being
different from the integer case.
In a companion paper, we use these results to characterize the limiting
behavior of partial sums of multiplicative functions in function fields and in
particular to solve a "corrected" form of the Erd\H{o}s discrepancy problem
over Fq[t].Comment: 62 pages; further referee comments incorporated; to appear in
Mathematik