Improved Helly numbers of product sets

A finite family $\mathcal F$ of convex sets is $k$-intersecting in $S \subseteq \mathbb{R}^d$ if the intersection of every subset of $k$ convex sets in $\mathcal F$ contains a point in $S$. The Helly number of $S$ is the minimum $k$, if it exists, such that every $k$-intersecting family contains a point of $S$ in its intersection. In this paper, we improve bounds on the Helly number of product sets of the form $A^d$ for various sets $A \subseteq \mathbb{R}$, including the ``exponential grid'' $A = \{\alpha^n : n \in \mathbb{N}\}$ and sets $A\subseteq \mathbb{Z}$ defined by congruence relations.