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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
DOI: https://doi.org/10.1090/S0002-9939-04-07344-7
Published electronically: June 2, 2004
MathSciNet review: 2063100
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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.


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Additional Information

Julien Cassaigne
Affiliation: Institut de Mathématiques de Luminy, 163 avenue de Luminy, Case 907, F-13288 Marseille Cedex 9, France
Email: cassaigne@iml.univ-mrs.fr

Alain Plagne
Affiliation: CMAT, École polytechnique, F-91128 Palaiseau Cedex, France
Email: plagne@math.polytechnique.fr

DOI: https://doi.org/10.1090/S0002-9939-04-07344-7
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