Correct order on some certain weighted representation functions

Chen, Shi-Qiang; Ding, Yuchen; Lü, Xiaodong; Zhang, Yuhan

Comptes Rendus. Mathématique, Tome 362 (2024) no. G5, pp. 547-552

Let $\mathbb{N}$ be the set of all nonnegative integers. For any positive integer $k$ and any subset $A$ of nonnegative integers, let $r_{1,k}(A,n)$ be the number of solutions $(a_1,a_2)$ to the equation $n=a_1+k{a}_2$. In 2016, Qu proved that

$$\underset{n\to \infty }{lim\; inf}{r}_{1,k}(A,n)=\infty$$

providing that $r_{1,k}(A,n)=r_{1,k}(\mathbb{N}\setminus A,n)$ for all sufficiently large integers, which answered affirmatively a 2012 problem of Yang and Chen. In a very recent article, another Chen (the first named author) slightly improved Qu’s result and obtained that

$$\underset{n\to \infty }{lim\; inf}\frac{r_{1,k}(A,n)}{logn}>0.$$

In this note, we further improve the lower bound on $r_{1,k}(A,n)$ by showing that

$$\underset{n\to \infty }{lim\; inf}\frac{r_{1,k}(A,n)}{n}>0.$$

Our bound reflects the correct order of magnitude of the representation function $r_{1,k}(A,n)$ under the above restrictions due to the trivial fact that $r_{1,k}(A,n)\le n/k.$

Soit $\mathbb{N}$ l’ensemble de tous les entiers non négatifs. Pour tout entier positif $k$ et tout sous-ensemble $A$ d’entiers non négatifs, notons $r_{1,k}(A,n)$ le nombre de solutions $(a_1,a_2)$ de l’équation $n=a_1+k{a}_2$. En 2016, Qu a prouvé que

$$\underset{n\to \infty }{lim\; inf}{r}_{1,k}(A,n)=\infty$$

ce qui signifie que $r_{1,k}(A,n)=r_{1,k}(\mathbb{N}\setminus A,n)$ pour tous les entiers suffisamment grands, ce qui répondait par l’affirmative à un problème de Yang et Chen datant de 2012. Dans un article très récent, un autre Chen (le premier auteur dans notre article) a légèrement amélioré le résultat de Qu et obtenu que

$$\underset{n\to \infty }{lim\; inf}\frac{r_{1,k}(A,n)}{logn}>0.$$

Dans cette note, nous améliorons encore le minorant de $r_{1,k}(A,n)$ en montrant que

$$\underset{n\to \infty }{lim\; inf}\frac{r_{1,k}(A,n)}{n}>0.$$

Notre limite reflète l’ordre de grandeur correct de la fonction de représentation $r_{1,k}(A,n)$ sous les restrictions ci-dessus en raison du fait trivial que $r_{1,k}(A,n)\le n/k$.

Classification : 11B34, 11A41
Keywords: representation functions, order of functions, partitions of integers
Mots-clés : fonctions de représentation, ordre des fonctions, partitions d’entiers

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