  We study deviation of ergodic averages for dynamical systems given by
self-similar tilings on the plane and in higher dimensions. The main object of
our paper is a special family of finitely-additive measures for our systems. An
asymptotic formula is given for ergodic integrals in terms of these
finitely-additive measures, and, as a corollary, limit theorems are obtained
for dynamical systems given by self-similar tilings.
