Large subsets of discrete hypersurfaces in Z d contain arbitrarily many collinear points

In 1977 L.T. Ramsey showed that any sequence in 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 N , let f : Z d Z d + 1 be a Lipschitz map and let A 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.