Linear equations over multiplicative groups in positive characteristic, sums of recurrences, and ergodic mixing

Solving completely $x+y-z=1$ in unknowns taken from the group generated by a variable $t$ with $1-t$ over a finite field is not so easy as might be expected. We present a generalization to arbitrary linear varieties and finitely generated groups (keywords effective Mordell-Lang). We also mention applications to (a) solving equations like $u_n+v_m+w_l+f_k=0$ in $n,m,l,k$ for given recurrences $u,v,w,f$; and to (b) finding the smallest order of non-mixing of a given algebraic ${\bf Z}^s$-action. This is joint work with Harm Derksen.