Abstract

We prove a pointwise convergence result for additive ergodic averages associated with certain multiplicative actions of the Gaussian integers. We derive several applications in dynamics and number theory, including: (i) Wirsing's theorem for Gaussian integers: if f : G-+ R is a bounded completely multiplicative function, then the following limit exists: lim N-).oo 1 Sigma N 2 1G m,n G N f ( m + i n ) . (ii) An answer to a special case of a question of Frantzikinakis and Host: for any completely multiplicative real-valued function f : N-+ R, the following limit exists: lim N-).oo 1 Sigma N 2 1G m,n G N f ( m 2 + n 2 ) . (iii) A variant of a theorem of Bergelson and Richter on ergodic averages along the Omega function: if (X, X, T ) is a uniquely ergodic system with unique invariant measure mu , then for any x is an element of X and f is an element of C ( X ), lim N-).oo 1 Sigma N 2 1G m,n G N f ( T Omega( m 2 + n 2 ) x ) = f d mu. X