Primitive Averages, Directional Expansivity, and Quantitative Twisted Recurrence for Ergodic ${\mathbb{Z}}^d$-Actions

Abstract

We prove two new results about probability preserving actions $T:{\mathbb{Z}}^d\curvearrowright (X,\hat{I}¼)$. First, for a function $f\in L^2(\hat{I}¼)$, we provide an explicit formula for the $L^2(\hat{I}¼)$-limit of the average \[\frac{1}{|Q_N^\mathcal{P}|}\sum_{v \in Q_N^\mathcal{P}} T_v f\] where $\mathcal{P}\subset {\mathbb{Z}}^d$ is the set of primitive vectors, i.e. those for which the greatest common divisor of its components is $1$, and $Q_N^{\mathcal{P}}=[-N,N{]}^d\cap \mathcal{P}$. Second, for a set $A\subset X$ with $\hat{I}¼(A)>0$, we provide a spectral condition under which the set of $\epsilon$-expansive directions \[\left\{ v\in \mathbb{Z}^d \, : \, μ\left(\bigcup_{n\in \mathbb{Z}} T_{nv}A\right)>1-\varepsilon\right\}\] has lower density very close to $1$. As an application of our techniques we are also able to prove a quantitative variant of a twisted multiple recurrence theorem of Björklund, Fish and the first author (arXiv:2503.02501).