Ergodic Averages over Shrinking Sectors of $\mathbb Z[i]$

Abstract

We study ergodic averages over Gaussian integers in shrinking angular sectors, which impose an increasingly restrictive directional relation between the real and imaginary coordinates. In uniquely ergodic systems, averages of continuous functions along $Ω$ converge to the invariant integral uniformly in the base point and sector location for sectors whose widths shrink subpolynomially in the norm cutoff. For finitely generated multiplicative actions, we prove a quantitative sector--disk comparison in an explicit logarithmic shrinking range, with a power-saving error in $\log N$, uniformly in the base point and sector location. The exponents depend only on the number of distinct prime transformations, and the result requires neither ergodicity nor angular-distribution hypotheses on the sets of primes inducing the individual transformations. Strong unique ergodicity then yields convergence to the invariant integral at each fixed base point. Consequences include asymptotic independence of relative angular position and $T^{Ω(n)} x$, Gaussian Liouville cancellation, and joint equidistribution modulo integers of prime-factor counts associated with two prime classes having divergent reciprocal-norm sums.