TY - GEN

T1 - On Helly Numbers of Exponential Lattices

AU - Ambrus, Gergely

AU - Balko, Martin

AU - Frankl, Nóra

AU - Jung, Attila

AU - Naszódi, Márton

N1 - Publisher Copyright:
© Gergely Ambrus, Martin Balko, Nóra Frankl, Attila Jung, and Márton Naszódi; licensed under Creative Commons License CC-BY 4.0.

PY - 2023/6/1

Y1 - 2023/6/1

N2 - Given a set S ⊆ R2, define the 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 F of convex sets in R2 such that the intersection of any N or fewer members of F contains at least one point of S, there is a point of S common to all members of F. We prove that the Helly numbers of exponential lattices {αn : n ∈ N0}2 are finite for every α > 1 and we determine their exact values in some instances. In particular, we obtain H({2n : n ∈ N0}2) = 5, solving a problem posed by Dillon (2021). For real numbers α, β > 1, we also fully characterize exponential lattices L(α, β) = {αn : n ∈ N0} × {βn : n ∈ N0} with finite Helly numbers by showing that H(L(α, β)) is finite if and only if logα(β) is rational.

AB - Given a set S ⊆ R2, define the 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 F of convex sets in R2 such that the intersection of any N or fewer members of F contains at least one point of S, there is a point of S common to all members of F. We prove that the Helly numbers of exponential lattices {αn : n ∈ N0}2 are finite for every α > 1 and we determine their exact values in some instances. In particular, we obtain H({2n : n ∈ N0}2) = 5, solving a problem posed by Dillon (2021). For real numbers α, β > 1, we also fully characterize exponential lattices L(α, β) = {αn : n ∈ N0} × {βn : n ∈ N0} with finite Helly numbers by showing that H(L(α, β)) is finite if and only if logα(β) is rational.

KW - Diophantine approximation

KW - exponential lattices

KW - Helly numbers

UR - http://www.scopus.com/inward/record.url?scp=85163548614&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SoCG.2023.8

DO - 10.4230/LIPIcs.SoCG.2023.8

M3 - Conference contribution

AN - SCOPUS:85163548614

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 39th International Symposium on Computational Geometry, SoCG 2023

A2 - Chambers, Erin W.

A2 - Gudmundsson, Joachim

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 39th International Symposium on Computational Geometry, SoCG 2023

Y2 - 12 June 2023 through 15 June 2023

ER -