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On Bohman's conjecture related to a sum packing problem of Erdos


Authors: R. Ahlswede, H. Aydinian and L. H. Khachatrian
Journal: Proc. Amer. Math. Soc. 132 (2004), 1257-1265
MSC (2000): Primary 11P99; Secondary 05D05
DOI: https://doi.org/10.1090/S0002-9939-03-07188-0
Published electronically: September 29, 2003
MathSciNet review: 2053329
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Abstract: Motivated by a sum packing problem of Erdos, Bohman discussed an extremal geometric problem which seems to have an independent interest. Let $H$ be a hyperplane in $\mathbb R^n$ such that $H\cap\{0,\pm1\}^n=\{0^n\}$. The problem is to determine

\begin{displaymath}f(n)\triangleq\max_H\vert H\cap\{0,\pm1,\pm2\}^n\vert.\end{displaymath}

Bohman (1996) conjectured that

\begin{displaymath}f(n)=\frac 12 (1+\sqrt2)^n+\frac 12 (1-\sqrt2)^n.\end{displaymath}

We show that for some constants $c_1,c_2$ we have $c_1(2,538)^n<f(n)< c_2(2,723)^n$--disproving the conjecture. We also consider a more general question of the estimation of $\vert H\cap\{0,\pm1,\dots,\pm m\}\vert$, when $H\cap\{0,\pm1,\dots,\pm k\}=\{0^n\}$, $m>k>1$.


References [Enhancements On Off] (What's this?)

  • 1. T. Bohman, A sum packing problem of Erdos and the Conway-Guy sequence, Proc. Amer. Math. Soc., Vol. 124, No. 12, 3627-3636, 1996. MR 97b:11027
  • 2. P. Erdos, Problems and results in additive number theory, Colloque sur la Théorie des Nombres, Bruxelles, 1955, Liège; Masson and Cie, Paris, 1956. MR 18:18a
  • 3. J. H. Conway and R. K. Guy, Sets of natural numbers with distinct sums, Notices Amer. Math. Soc., Vol. 15, p. 345, 1968.

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Additional Information

R. Ahlswede
Affiliation: Department of Mathematics, University of Bielefeld, POB 100131, 33501 Bielefeld, Germany
Email: ahlswede@mathematik.uni-bielefeld.de

H. Aydinian
Affiliation: Department of Mathematics, University of Bielefeld, POB 100131, 33501 Bielefeld, Germany
Email: ayd@mathematik.uni-bielefeld.de

L. H. Khachatrian
Affiliation: Department of Mathematics, University of Bielefeld, POB 100131, 33501 Bielefeld, Germany
Email: lk@mathematik.uni-bielefeld.de

DOI: https://doi.org/10.1090/S0002-9939-03-07188-0
Received by editor(s): October 22, 2001
Received by editor(s) in revised form: August 22, 2002, and January 15, 2003
Published electronically: September 29, 2003
Communicated by: John R. Stembridge
Article copyright: © Copyright 2003 American Mathematical Society

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