Research article - Combinatorics, number theory
Correct order on some certain weighted representation functions
Comptes Rendus. Mathématique, Volume 362 (2024), pp. 547-552.

Let $ℕ$ be the set of all nonnegative integers. For any positive integer $k$ and any subset $A$ of nonnegative integers, let ${r}_{1,k}\left(A,n\right)$ be the number of solutions $\left({a}_{1},{a}_{2}\right)$ to the equation $n={a}_{1}+k{a}_{2}$. In 2016, Qu proved that

 $\underset{n\phantom{\rule{0.166667em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}\infty }{lim inf}{r}_{1,k}\left(A,n\right)=\infty$

providing that ${r}_{1,k}\left(A,n\right)={r}_{1,k}\left(ℕ\setminus A,n\right)$ 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

 $\underset{n\phantom{\rule{0.166667em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}\infty }{lim inf}\frac{{r}_{1,k}\left(A,n\right)}{logn}>0.$

In this note, we further improve the lower bound on ${r}_{1,k}\left(A,n\right)$ by showing that

 $\underset{n\phantom{\rule{0.166667em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}\infty }{lim inf}\frac{{r}_{1,k}\left(A,n\right)}{n}>0.$

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

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 ${r}_{1,k}\left(A,n\right)$ le nombre de solutions $\left({a}_{1},{a}_{2}\right)$ de l’équation $n={a}_{1}+k{a}_{2}$. En 2016, Qu a prouvé que

 $\underset{n\phantom{\rule{0.166667em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}\infty }{lim inf}{r}_{1,k}\left(A,n\right)=\infty$

ce qui signifie que ${r}_{1,k}\left(A,n\right)={r}_{1,k}\left(ℕ\setminus A,n\right)$ 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

 $\underset{n\phantom{\rule{0.166667em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}\infty }{lim inf}\frac{{r}_{1,k}\left(A,n\right)}{logn}>0.$

Dans cette note, nous améliorons encore le minorant de ${r}_{1,k}\left(A,n\right)$ en montrant que

 $\underset{n\phantom{\rule{0.166667em}{0ex}}\to \phantom{\rule{0.166667em}{0ex}}\infty }{lim inf}\frac{{r}_{1,k}\left(A,n\right)}{n}>0.$

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

Revised:
Accepted:
Published online:
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

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
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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/

[1] J. P. Bell; J. Shallit Counterexamples to a conjecture of Dombi in additive number theory, Acta Math. Hung., Volume 169 (2023), pp. 562-565 | DOI | MR | Zbl

[2] S.-Q. Chen The lower bound of weighted representation function (preprint), to appear in Period. Math. Hungar | arXiv

[3] Y.-G. Chen On the values of representation functions, Sci. China, Math., Volume 54 (2011) no. 7, pp. 1317-1331 | DOI | MR | Zbl

[4] Y.-G. Chen; M. Tang Partitions of natural numbers with the same representation functions, J. Number Theory, Volume 129 (2009), pp. 2689-2695 | DOI | MR | Zbl

[5] Y.-G. Chen; B. Wang On additive properties of two special sequences, Acta Arith., Volume 110 (2003), pp. 299-303 | DOI | MR | Zbl

[6] G. Dombi Additive properties of certain sets, Acta Arith., Volume 103 (2002), pp. 137-146 | DOI | MR | Zbl

[7] V. F. Lev Reconstructing integer sets from their representation functions, Electron. J. Comb., Volume 11 (2004) no. 1, R78 | MR | Zbl

[8] Y.-L. Li; W.-X. Ma Partitions of natural numbers with the same weighted representation functions, Colloq. Math., Volume 159 (2020), pp. 1-5 | DOI | MR | Zbl

[9] Z. Qu A note on representation functions with different weights, Colloq. Math., Volume 143 (2016), pp. 105-112 | MR | Zbl

[10] J. Shallit A Dombi counterexample with positive lower density, Integers, Volume 23 (2023), #A74 | DOI | MR

[11] A. Sárközy; V. T. Sós On additive representation functions, The mathematics of Paul Erdős. Vol. I. (Ronald L. Graham et al., eds.) (Algorithms and Combinatorics), Volume 13, Springer, 1997, pp. 129-150 | DOI | MR | Zbl

[12] C. Sándor Partitions of natural numbers and their representation functions, Integers, Volume 4 (2004), A18 | MR | Zbl

[13] M. Tang Partitions of the set of natural numbers and their representation functions, Discrete Math., Volume 308 (2008), pp. 2614-2616 | DOI | MR | Zbl

[14] Q.-H. Yang Representation functions with different weights, Colloq. Math., Volume 137 (2014), pp. 1-6 | DOI | MR | Zbl

[15] Q.-H. Yang; Y.-G. Chen Partitions of natural numbers with the same weighted representation functions, J. Number Theory, Volume 132 (2012), pp. 3047-3055 | DOI | MR | Zbl

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