Georgia Institute of Technology College of Sciences

We are going to talk about  ergodic averages over Gaussian integers in angular sectors whose widths shrink with the norm cutoff. For finitely generated multiplicative actions, we prove a uniform sector–disk comparison with an explicit logarithmic shrinking range and a power-saving error, without assuming ergodicity or angular distribution of the underlying prime transformations. We then apply strong unique ergodicity to obtain convergence to the invariant integral. Our results also yield asymptotic independence of angular position and the orbit, Gaussian Liouville cancellation, and joint equidistribution of prime-factor counts associated with distinct prime classes.