On Helly numbers of exponential lattices
| Autor: | Ambrus, Gergely, Balko, Martin, Frankl, Nóra, Jung, Attila, Naszódi, Márton |
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| Jazyk: | angličtina |
| Rok vydání: | 2023 |
| Předmět: |
Mathematics - Metric Geometry
Mathematics - Number Theory Diophantine approximation exponential lattices FOS: Mathematics Mathematics - Combinatorics Metric Geometry (math.MG) Combinatorics (math.CO) Number Theory (math.NT) Helly numbers Mathematics of computing → Combinatoric problems 52A35 52A38 52C07 |
| Popis: | 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. Comment: 24 pages, 5 figures, minor changes |
| Databáze: | OpenAIRE |
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