Grekos’ S function has a linear growth

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 $h$): \[ S (h) = \max _{\mathcal {A}} \qquad \limsup _{a \in \mathcal {A}^{\ast }} \qquad \mbox {ord}^{\ast } ( \mathcal {A} \setminus \{ a \}). \] 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.