TY - JOUR
T1 - Large subsets of discrete hypersurfaces in Zd contain arbitrarily many collinear points
AU - Moreira, Joel
AU - Richter, Florian Karl
N1 - Publisher Copyright:
© 2016.
PY - 2016/5/1
Y1 - 2016/5/1
N2 - In 1977 L.T. Ramsey showed that any sequence in Z2 with bounded gaps contains arbitrarily many collinear points. Thereafter, in 1980, C. Pomerance provided a density version of this result, relaxing the condition on the sequence from having bounded gaps to having gaps bounded on average.We give a higher dimensional generalization of these results. Our main theorem is the following. Theorem. Let d∈N, let f:Zd→Zd+1 be a Lipschitz map and let A⊂Zd have positive upper Banach density. Then f(A) contains arbitrarily many collinear points.Note that Pomerance's theorem corresponds to the special case d= 1. In our proof, we transfer the problem from a discrete to a continuous setting, allowing us to take advantage of analytic and measure theoretic tools such as Rademacher's theorem.
AB - In 1977 L.T. Ramsey showed that any sequence in Z2 with bounded gaps contains arbitrarily many collinear points. Thereafter, in 1980, C. Pomerance provided a density version of this result, relaxing the condition on the sequence from having bounded gaps to having gaps bounded on average.We give a higher dimensional generalization of these results. Our main theorem is the following. Theorem. Let d∈N, let f:Zd→Zd+1 be a Lipschitz map and let A⊂Zd have positive upper Banach density. Then f(A) contains arbitrarily many collinear points.Note that Pomerance's theorem corresponds to the special case d= 1. In our proof, we transfer the problem from a discrete to a continuous setting, allowing us to take advantage of analytic and measure theoretic tools such as Rademacher's theorem.
UR - http://www.scopus.com/inward/record.url?scp=84954224329&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84954224329&partnerID=8YFLogxK
U2 - 10.1016/j.ejc.2015.12.012
DO - 10.1016/j.ejc.2015.12.012
M3 - Article
AN - SCOPUS:84954224329
SN - 0195-6698
VL - 54
SP - 163
EP - 176
JO - European Journal of Combinatorics
JF - European Journal of Combinatorics
ER -