On Helly numbers of exponential lattices

Abstract

Given a set $S\subseteq {\mathbb{R}}^2$, define the \emph{Helly number of $S$}, denoted by $H(S)$, as the smallest positive integer $N$, if it exists, for which the following statement is true: for any finite family $\mathcal{F}$ of convex sets in~${\mathbb{R}}^2$ such that the intersection of any $N$ or fewer members of~$\mathcal{F}$ contains at least one point of $S$, there is a point of $S$ common to all members of $\mathcal{F}$. We prove that the Helly numbers of \emph{exponential lattices} $\{{\alpha }^n:n\in {\mathbb{N}}_0{\}}^2$ are finite for every $\alpha >1$ and we determine their exact values in some instances. In particular, we obtain $H(\{2^n:n\in {\mathbb{N}}_0{\}}^2)=5$, solving a problem posed by Dillon (2021). For real numbers $\alpha ,\beta >1$, we also fully characterize exponential lattices $L(\alpha ,\beta )=\{{\alpha }^n:n\in {\mathbb{N}}_0\}\times \{{\beta }^n:n\in {\mathbb{N}}_0\}$ with finite Helly numbers by showing that $H(L(\alpha ,\beta ))$ is finite if and only if ${\log }_{\alpha }(\beta )$ is rational.

24 pages, 5 figures, minor changes