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 f ( z ) n f ( z + c ) assumes every non-zero value a ∈ C infinitely often. This research also shows that there exist two sets S 1 with 9 (resp. 5) elements and S 2 with 1 element, such that for a finite order nonconstant meromorphic (resp. entire) function f and a non-zero complex constant c, E f ( z ) ( S j ) = E f ( z + c ) ( S j ) ( 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.