Discrete Analogues in Harmonic Analysis: Directional Maximal Functions in ℤ2

Abstract Let $V = \{ v_1,\dots ,v_N\}\subset \mathbb{Z}^2$ be a collection of $N$ vectors that lie near a discrete sphere. We consider discrete directional maximal functions on $\mathbb{Z}^2$ where the set of directions lies in $V$, given by $$\begin{align*}& \sup_{v \in V, k \geq C \log N} \left| \sum_{n \in \mathbb{Z}} f(x-nv ) \phi_k(n) \right|, \ f:\mathbb{Z}^2 \to \mathbb{C},\end{align*}$$where $\phi _k(t):= 2^{-k} \phi (2^{-k} t)$ for some bump function $\phi$. Interestingly, the study of these operators leads one to consider an “arithmetic version” of a Kakeya-type problem in the plane, which we approach using a combination of geometric and number-theoretic methods. Motivated by the Furstenberg problem from geometric measure theory, we also consider a discrete directional maximal operator along polynomial orbits, $$\begin{align*}& \sup_{v \in V} \left| \sum_{n \in \mathbb{Z}} f(x-v\cdot P(n) ) \cdot \phi(n) \right|, \ P \in \mathbb{Z}[-].\end{align*}$$