For $\vec{d} \in \mathbb{N}^n$, let $Q(\vec{d}) \subset \mathbb{N}^n$ be the set of vertices of the $n$-dimensional cube scaled in the direction of the $i$-th coordinate by $d_i$, i.e. $Q(\vec{d} = \{\langle \pm d_1, \ldots, \pm d_n\rangle\}$.
Consider the following problem:
Given a set of points in $\mathbb{N}^n$ and number $k$, does the set contains an $n$-dimentional arithmetic progression of length $k$?
More formally,
Input:
given a finite set $X \subseteq \mathbb{N}^n$ and a positive integer $k \in \mathbb{N}^+$.Question:
are there $\vec{o}\in \mathbb{N}^n$ and $ \vec{d} \in (\mathbb{N}^+)^n$ such that $\vec{o}+ Q(i\vec{d})\subseteq X$ for all integers $0 \leq i \leq k$?
Informally we are looking at the containment of the vertices of scaled $n$-dimensional axis-aligned cubes centered at $\vec{o}$.
Does this problem have a name? What is its complexity? Can we solved it using dynamic programming?