Large subsets of discrete hypersurfaces in $\mathbb{Z}^d$ contain arbitrarily many collinear points

TL;DR

This paper generalizes classical results on collinear points in sequences to higher dimensions, showing that large dense subsets of integer lattices mapped Lipschitz-continuously contain arbitrarily many collinear points.

Contribution

It extends Pomerance's density result from one dimension to higher dimensions using a measure-theoretic approach.

Findings

Higher-dimensional generalization of collinearity in dense subsets

Lipschitz maps preserve collinear structures in dense sets

Use of measure theory to analyze discrete geometric problems

Abstract

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.