  In 1977 L.T. Ramsey showed that any sequence in $\mathbb{Z}^2$ with bounded
gaps contains arbitrarily many collinear points. Thereafter, in 1980, C.
Pomerance provided a density version of this result, relaxing the condition on
the sequence from having bounded gaps to having gaps bounded on average. We
give a higher dimensional generalization of these results. Our main theorem is
the following.
  Theorem: Let $d\in\mathbb{N}$, let $f:\mathbb{Z}^d\to\mathbb{Z}^{d+1}$ be a
Lipschitz map and let $A\subset\mathbb{Z}^d$ have positive upper Banach
density. Then $f(A)$ contains arbitrarily many collinear points.
  Note that Pomerance's theorem corresponds to the special case $d=1$. In our
proof, we transfer the problem from a discrete to a continuous setting,
allowing us to take advantage of analytic and measure theoretic tools such as
Rademacher's theorem.
