Result: A lattice point problem and additive number theory

Title:
A lattice point problem and additive number theory
Authors:
Moshe Dubiner, Noga Alon
Source:
Combinatorica. 15:301-309
Publisher Information:
Springer Science and Business Media LLC, 1995.
Publication Year:
1995
Subject Terms:
Additive number theory, partitions, Other combinatorial number theory, eigenvalues of Cayley graphs, 0102 computer and information sciences, 0101 mathematics, 01 natural sciences, lattice points
Document Type:
Academic journal Article
File Description:
application/xml
Language:
English
ISSN:
1439-6912
0209-9683
DOI:
10.1007/bf01299737
Access URL:
http://www.math.tau.ac.il/~nogaa/PDFS/centroid.pdf
https://dblp.uni-trier.de/db/journals/combinatorica/combinatorica15.html#AlonD95
https://doi.org/10.1007/BF01299737
https://collaborate.princeton.edu/en/publications/a-lattice-point-problem-and-additive-number-theory
https://link.springer.com/content/pdf/10.1007/BF01299737.pdf
Rights:
Springer TDM
Accession Number:
edsair.doi.dedup.....06259dfaab67c557cf807018570f0b46
Database:
OpenAIRE

Further Information

Let \(f(n,d)\) denote the minimal number such that every set of \(f(n,d)\) lattice points in a \(d\) dimensional space contains \(n\) points whose centroid (mean) is also a lattice point. Here the estimate \(f(n,d) < (cd \log d)^dn\) is proved, a sharp one for fixed \(d\) and \(n \to \infty\). The proof combines Plünnecke's ideas, expansion properties of graphs with given eigenvalues, and classical exponential sums to evaluate the eigenvalues of certain Cayley graphs.