  Certain estimates involving the derivative $f\mapsto f'$ of a meromorphic
function play key roles in the construction and applications of classical
Nevanlinna theory. The purpose of this study is to extend the usual Nevanlinna
theory to a theory for the exact difference $f\mapsto \Delta f=f(z+c)-f(z)$.
  An $a$-point of a meromorphic function $f$ is said to be $c$-paired at
$z\in\C$ if $f(z)=a=f(z+c)$ for a fixed constant $c\in\C$. In this paper the
distribution of paired points of finite-order meromorphic functions is studied.
One of the main results is an analogue of the second main theorem of Nevanlinna
theory, where the usual ramification term is replaced by a quantity expressed
in terms of the number of paired points of $f$. Corollaries of the theorem
include analogues of the Nevanlinna defect relation, Picard's theorem and
Nevanlinna's five value theorem. Applications to difference equations are
discussed, and a number of examples illustrating the use and sharpness of the
results are given.
