Value distribution and shared sets of differences of meromorphic functions

We investigate value distribution and uniqueness problems of difference polynomials of meromorphic functions. In particular, we show that for a finite order transcendental meromorphic function f with λ(1/f)<ρ(f) and a non-zero complex constant c, if n⩾2, then fn(z)f(z+c) assumes every non-zero value a∈C infinitely often. This research also shows that there exist two sets S1 with 9 (resp. 5) elements and S2 with 1 element, such that for a finite order nonconstant meromorphic (resp. entire) function f and a non-zero complex constant c, Ef(z)(Sj)=Ef(z+c)(Sj) (j=1,2) imply f(z)≡f(z+c). This gives an answer to a question of Gross concerning a finite order meromorphic function f and its shift.