On Helly numbers of exponential lattices

TL;DR

This paper investigates the Helly numbers of exponential lattices in the plane, proving their finiteness for all bases greater than one and explicitly determining these values in certain cases, including a solved problem by Dillon.

Contribution

It establishes the finiteness of Helly numbers for exponential lattices and characterizes when these numbers are finite based on the rationality of the logarithmic ratio.

Findings

Helly numbers are finite for all exponential lattices with base > 1.

Exact Helly number for the lattice generated by powers of 2 is 5.

Helly number is finite if and only if the logarithmic ratio of bases is rational.

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} $\{\alpha^n \colon n \in \mathbb{N}_0\}^2$ are finite for every $\alpha>1$ and we determine their exact values in some instances. In particular, we obtain $H(\{2^n \colon n \in \mathbb{N}_0\}^2)=5$, solving a problem posed by Dillon (2021). For real numbers $\alpha, \beta > 1$, we also fully characterize exponential lattices $L(\alpha,\beta) = \{\alpha^n \colon n \in \mathbb{N}_0\} \times \{\beta^n \colon n \in \mathbb{N}_0\}$ with finite Helly numbers by showing that $H(L(\alpha,\beta))$ is finite if and only if $\log_\alpha(\beta)$ is rational.