Correct order on some certain weighted representation functions

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

doi:10.5802/crmath.573

Article de recherche - Combinatoire, théorie des nombres

Correct order on some certain weighted representation functions
[L’ordre correct de certaines fonctions de représentation pondérées]

Shi-Qiang Chen ¹ ; Yuchen Ding ² ; Xiaodong Lü ² ; Yuhan Zhang ²

¹ School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, People’s Republic of China
² School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, People’s Republic of China

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 547-552.

Résumé
Abstract

Soit $ℕ$ 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 $r_{1, k} (A, n) = r_{1, k} (ℕ ∖ A, n)$ 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)}{log n} > 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 $r_{1, k} (A, n) \leq n / k$ .

Let $ℕ$ 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 $r_{1, k} (A, n) = r_{1, k} (ℕ ∖ A, n)$ 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)}{log n} > 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 $r_{1, k} (A, n) \leq n / k .$

Reçu le : 2023-06-29
Révisé le : 2023-08-17
Accepté le : 2023-09-19
Publié le : 2024-05-31

DOI : 10.5802/crmath.573

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

Affiliations des auteurs :

Shi-Qiang Chen ¹ ; Yuchen Ding ² ; Xiaodong Lü ² ; Yuhan Zhang ²

¹ School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, People’s Republic of China
² School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, People’s Republic of China

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Shi-Qiang Chen and Yuchen Ding and Xiaodong L\"u and Yuhan Zhang},
     title = {Correct order on some certain weighted representation functions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {547--552},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.573},
     language = {en},
}

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Shi-Qiang Chen; Yuchen Ding; Xiaodong Lü; Yuhan Zhang. Correct order on some certain weighted representation functions. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 547-552. doi : 10.5802/crmath.573. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.573/

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