Tverberg theorems over discrete sets of points

This paper discusses Tverberg-type theorems with coordinate constraints (i.e., versions of these theorems where all points lie within a subset \(S \subset \mathbb{R}^d\) and the intersection of convex hulls is required to have a non-empty intersection with \(S\)). We determine the \(m\)-Tverberg number, when \(m \geq 3\), of any discrete subset \(S\) of \(\mathbb{R}^2\) (a generalization of an unpublished result of J.-P. Doignon). We also present improvements on the upper bounds for the Tverberg numbers of \(\mathbb{Z}^3\) and \(\mathbb{Z}^j \times \mathbb{R}^k\) and an integer version of the well-known positive-fraction selection lemma of J. Pach.