Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Grekos' S function has a linear growth

Authors: Julien Cassaigne and Alain Plagne
Journal: Proc. Amer. Math. Soc. 132 (2004), 2833-2840
MSC (2000): Primary 11B13
Published electronically: June 2, 2004
MathSciNet review: 2063100
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An exact additive asymptotic basis is a set of nonnegative integers such that there exists an integer $h$ with the property that any sufficiently large integer can be written as a sum of exactly $h$ elements of $\mathcal{A}$. The minimal such $h$ is the exact order of $\mathcal{A}$ (denoted by $ \mbox{ord}^{\ast} ( \mathcal{A} )$). Given any exact additive asymptotic basis $\mathcal{A}$, we define $\mathcal{A}^{\ast}$ to be the subset of $\mathcal{A}$ composed with the elements $a \in \mathcal{A}$ such that $\mathcal{A} \setminus \{ a \}$ is still an exact additive asymptotic basis. It is known that $ \mathcal{A} \setminus \mathcal{A}^{\ast}$ is finite.

In this framework, a central quantity introduced by Grekos is the function $S(h)$ defined as the following maximum (taken over all bases $\mathcal{A}$ of exact order $h$):

\begin{displaymath}S (h) = \max_{\mathcal{A}} \hspace*{.1in} \limsup_{a \in \mat... ...ace*{.1in} \mbox{ord}^{\ast} ( \mathcal{A} \setminus \{ a \}). \end{displaymath}

In this paper, we introduce a new and simple method for the study of this function. We obtain a new estimate from above for $S$ which improves drastically and in any case on all previously known estimates. Our estimate, namely $S(h) \leq 2h$, cannot be too far from the truth since $S$ verifies $S(h) \geq h+1$. However, it is certainly not always optimal since $S(2)=3$. Our last result shows that $S (h)$ is in fact a strictly increasing sequence.

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

  • 1. R. de la Bretèche, Problèmes extrémaux pour les bases additives, manuscript (2001).
  • 2. Bruno Deschamps and Georges Grekos, Estimation du nombre d’exceptions à ce qu’un ensemble de base privé d’un point reste un ensemble de base, J. Reine Angew. Math. 539 (2001), 45–53 (French, with English summary). MR 1863853,
  • 3. P. Erdos, Einige Bemerkungen zur Arbeit von A. Stöhr ``Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe'', J. reine angew. Math. 197 (1957), 216-219. MR 19:122b
  • 4. P. Erdős and R. L. Graham, On bases with an exact order, Acta Arith. 37 (1980), 201–207. MR 598875
  • 5. P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory, Monographies de L’Enseignement Mathématique [Monographs of L’Enseignement Mathématique], vol. 28, Université de Genève, L’Enseignement Mathématique, Geneva, 1980. MR 592420
  • 6. George P. Grekos, Minimal additive bases and related problems, Number theory days, 1980 (Exeter, 1980) London Math. Soc. Lecture Note Ser., vol. 56, Cambridge Univ. Press, Cambridge, 1982, pp. 300–305. MR 697273
  • 7. Georges Grekos, Sur l’ordre d’une base additive, Séminaire de Théorie des Nombres, 1987–1988 (Talence, 1987–1988) Univ. Bordeaux I, Talence, 198?, pp. Exp. No. 31, 13 (French). MR 993125
  • 8. Georges Grekos, Extremal problems about additive bases, Proceedings of the 13th Czech and Slovak International Conference on Number Theory (Ostravice, 1997), 1998, pp. 87–92. MR 1822518
  • 9. H. Halberstam and K. F. Roth, Sequences. Vol. I, Clarendon Press, Oxford, 1966. MR 0210679
  • 10. E. Härtter, Ein Beitrag zur Theorie der Minimalbasen, J. reine angew. Math. 196 (1956), 170-204. MR 19:122a
  • 11. John C. M. Nash, Some applications of a theorem of M. Kneser, J. Number Theory 44 (1993), no. 1, 1–8. MR 1219479,
  • 12. Melvyn B. Nathanson, Minimal bases and powers of 2, Acta Arith. 49 (1988), no. 5, 525–532. MR 967335
  • 13. Alain Plagne, Removing one element from an exact additive basis, J. Number Theory 87 (2001), no. 2, 306–314. MR 1824151,
  • 14. A. Stöhr, Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe I, II, J. reine angew. Math. 194 (1955), 40-65 and 111-140. MR 17:713a

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11B13

Retrieve articles in all journals with MSC (2000): 11B13

Additional Information

Julien Cassaigne
Affiliation: Institut de Mathématiques de Luminy, 163 avenue de Luminy, Case 907, F-13288 Marseille Cedex 9, France

Alain Plagne
Affiliation: CMAT, École polytechnique, F-91128 Palaiseau Cedex, France

Keywords: Additive basis, exact order
Received by editor(s): June 17, 2002
Published electronically: June 2, 2004
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2004 American Mathematical Society