Large subsets of discrete hypersurfaces in Zd contain arbitrarily many collinear points

In 1977 L.T. Ramsey showed that any sequence in Z2 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:Zd→Zd+1be a Lipschitz map and let  A⊂Zdhave positive upper Banach density. Then  f(A)f(A)contains arbitrarily many collinear points.Note that Pomerance’s theorem corresponds to the special case d=1d=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.