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 -