  We study correlation estimates of automatic sequences (that is, sequences
computable by finite automata) with polynomial phases. As a consequence, we
provide a new class of good weights for classical and polynomial ergodic
theorems, not coming themselves from dynamical systems.
  We show that automatic sequences are good weights in $L^2$ for polynomial
averages and totally ergodic systems. For totally balanced automatic sequences
(i.e., sequences converging to zero in mean along arithmetic progressions) the
pointwise weighted ergodic theorem in $L^1$ holds. Moreover, invertible
automatic sequences are good weights for the pointwise polynomial ergodic
theorem in $L^r$, $r>1$.
