A lattice point problem and additive number theory


For every dimensiond≥1 there exists a constantc=c(d) such that for alln≥1, every set of at leastcn lattice points in thed-dimensional Euclidean space contains a subset of cardinality preciselyn whose centroid is also a lattice point. The proof combines techniques from additive number theory with results about the expansion properties of Cayley graphs with given eigenvalues.

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  1. [1]

    N. Alon: Subset sums,J. Number Theory 27 (1987), 196–205.

    Google Scholar 

  2. [2]

    N. Alon, andM. Dubiner:Zero-sum sets of prescribed size, in: Combinatorics, Paul Erdös is Eighty, Bolyai Society, Mathematical Studies, Keszthely, Hungary, 1993, 33–50.

    Google Scholar 

  3. [3]

    N. Alon, R. M. Karp, D. Peleg, andD. B. West: A graph-theoretic game and its application to thek-servers problem,SIAM J. Comp., to appear.

  4. [4]

    N. Alon, andV. D. Milman: λ1, isoperimetric inequalities for graphs and superconcentrators,J. Comb. Theory, Ser. B,38 (1985), 73–88.

    Google Scholar 

  5. [5]

    N. Alon, andJ. H. Spencer:The probabilistic method, Wiley, 1991.

  6. [6]

    T. C. Brown andJ. C. Buhler: A density version of a geometric Ramsey theorem,J. Comb. Theory, Ser. A,32 (1982), 20–34.

    Google Scholar 

  7. [7]

    J. L. Brenner: Problem 6298,Amer. Math. Monthly,89 (1982), 279–280.

    Google Scholar 

  8. [8]

    P. Erdős, A. Ginzburg, andA. Ziv: Theorem in the additive number theory,Bull. Research Council Isreal 10F (1961), 41–43.

    Google Scholar 

  9. [9]

    P. Erdős andH. Heilbronn: On the addition of residue classes modp, Acta Arith.,9 (1964), 149–159.

    Google Scholar 

  10. [10]

    P. Frankl, R. L. Graham, andV. Rödl: On subsets of abelian groups with no 3-term arithmetic progression,J. Comb. Theory Ser. A,45 (1987), 157–161.

    Google Scholar 

  11. [11]

    H. Furstenberg, andY. Katznelson: An ergodic Szemerédi theorem for IP-systems and combinatorial theory,J. d'Analyse Math.,45 (1985), 117–168.

    Google Scholar 

  12. [12]

    H. Harborth: Ein Extremalproblem für Gitterpunkte,J. Reine Angew. Math.,262/263 (1973), 356–360.

    Google Scholar 

  13. [13]

    A. Kemnitz: On a lattice point problem,Ars Combinatoria,16b (1983), 151–160.

    Google Scholar 

  14. [14]

    B. Leeb, andC. Stahlke: A problem on lattice points,Crux Mathematicorum 13 (1987), 104–108.

    Google Scholar 

  15. [15]

    L. H. Loomis, andH. Whitney: An inequality related to the isoperimetric inequality,Bulletin AMS,55 (1949), 961–962.

    Google Scholar 

  16. [16]

    L. Lovász:Combinatorial Problems and Exercises, North Holland, Amsterdam, 1979, Problem 11.8.

    Google Scholar 

  17. [17]

    J. E. Olson: An addition theorem modulop, J. Comb. Theory 5 (1968), 45–52.

    Google Scholar 

  18. [18]

    H. Plünnecke: Eine zahlentheoretische Anwendung der Graphentheorie,J. Reine Angew. Math.,243 (1970), 171–183.

    Google Scholar 

  19. [19]

    I. Z. Ruzsa: An application of graph theory to additive number theory,Scientia,3 (1989), 97–109.

    Google Scholar 

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Alon, N., Dubiner, M. A lattice point problem and additive number theory. Combinatorica 15, 301–309 (1995). https://doi.org/10.1007/BF01299737

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Mathematics Subject Classification (1991)

  • 11 B 75