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

We prove two new results about probability preserving actions $T:{\mathbb{Z}}^d\curvearrowright (X,\mu )$. First, for a function $f\in L^2(\mu )$, we provide an explicit formula for the $L^2(\mu )$-limit of the average $$\frac{1}{\mathrm{\mid }{Q}_N^{\mathcal{P}}\mathrm{\mid }}\sum _{v\in Q_N^{\mathcal{P}}}{T}_{v}f$$ 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 $\mu (A)>0$, we provide a spectral condition under which the set of $\epsilon$-expansive directions $$\{v\in {\mathbb{Z}}^d:\mu \left(\bigcup _{n\in \mathbb{Z}}{T}_{nv}A\right)>1-\epsilon \}$$ 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).