Estimates for trilinear and quadrilinear character sums

We obtain new bounds on some trilinear and quadrilinear character sums, which are non-trivial starting from very short ranges of the variables. An application to an apparently new problem on oscillations of characters on differences between Farey fractions is given. Other applications include a modular analogue of a multiplicative hybrid problem of Iwaniec and Sárközy (1987) and the solvability of some prime type equations with constraints.