On Bohman’s conjecture related to a sum packing problem of Erdos
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- by R. Ahlswede, H. Aydinian and L. H. Khachatrian
- Proc. Amer. Math. Soc. 132 (2004), 1257-1265
- DOI: https://doi.org/10.1090/S0002-9939-03-07188-0
- Published electronically: September 29, 2003
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Abstract:
Motivated by a sum packing problem of Erdős, Bohman discussed an extremal geometric problem which seems to have an independent interest. Let $H$ be a hyperplane in $\mathbb R^n$ such that $H\cap \{0,\pm 1\}^n=\{0^n\}$. The problem is to determine \[ f(n)\triangleq \max _H|H\cap \{0,\pm 1,\pm 2\}^n|.\] Bohman (1996) conjectured that \[ f(n)=\frac 12 (1+\sqrt 2)^n+\frac 12 (1-\sqrt 2)^n.\] 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 $|H\cap \{0,\pm 1,\dots ,\pm m\}|$, when $H\cap \{0,\pm 1,\dots ,\pm k\}=\{0^n\}$, $m>k>1$.References
- Tom Bohman, A sum packing problem of Erdős and the Conway-Guy sequence, Proc. Amer. Math. Soc. 124 (1996), no. 12, 3627–3636. MR 1363448, DOI 10.1090/S0002-9939-96-03653-2
- Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
- J. H. Conway and R. K. Guy, Sets of natural numbers with distinct sums, Notices Amer. Math. Soc., Vol. 15, p. 345, 1968.
Bibliographic 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
- 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
- © Copyright 2003 American Mathematical Society
- 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
- MathSciNet review: 2053329