Partition Functions in Numeration Systems with Bounded Multiplicity

35 pagesInternational audienceFor a given increasing sequence of positive integers $A=(a_k)_{k\ge 0}$ and for $q$, an integer $\ge 2$ or eventually $q=\infty$, let $M_{A,q}(n)$ denote the number of representations of a given integer $n$ by sums $\sum_{k\ge 0} e_ka_k$ with integers $e_k$ in $[0,q)$. If $a_0=1$, the sequence $A$ constitutes a numeration system for the natural numbers and $A$ takes the name of scale. The partition problem consists in studying the asymptotic behavior of $M_{A,q}(\cdot)$ and its summation function $\Gamma_{A,q}(\cdot)$. In this paper we study various aspects of this problem. In the first part we recall important results and methods developed in the literature with attentions to the binary numeration system, the $d$-ary numeration system and also the Fibonacci and the $m$-bonacci scales. These cases show that $M_{A,q}(\cdot)$ can be very irregular. In the second part, miscellaneous general results are proved and we investigate in more details sequences $A$ which grow exponentially. In particular, we generalize a result of Dumont-Sidorov-Thomas in proving that if $a_k\sim c\gamma^k$ (with the only natural restriction $\gamma>1$) then $\Gamma_{A,q}(x)=x^{\log_\gamma q}H(\log_\gamma x)+o(x^{\log_\gamma q})$ where $H$ is a function strictly positive, continuous, periodic of period 1 and almost everywhere differentiable. The final part is devoted to a particular family of recurrent sequences $G$ called Pisot scales. We prove in that case that for any suitable $q$, there exists a set $S_{G,q}$ of positive integers with natural density 1 such that $\lim_{s\to\infty,\, s\in S_{G,q}} \log M_{G,q}(s)/\log s$ exists. The proof uses a previous work of D.-J. Feng and N. Sidorov related to the multiplicity of the radix $\theta$-expansions of real numbers using digits $0,1,\dots,q-1$