On two problems of Ljujić and Nathanson

Li-Xia Dai; Yong-Gao Chen

doi:10.1016/j.crma.2016.01.007

Number theory

On two problems of Ljujić and Nathanson

Presented by: the Editorial Board

Li-Xia Dai ¹; Yong-Gao Chen ¹

¹ School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, PR China

Comptes Rendus. Mathématique, Volume 354 (2016) no. 3, pp. 235-238.

Abstract
Résumé

Let N be the set of all nonnegative integers. For $A, M \subseteq N ∖ {0}$ and $n \in N$ , let $p (n, A, M)$ denote the number of representations of n in the form $n = \sum_{a \in A} m_{a} a$ , where $m_{a} \in M \cup {0}$ for all $a \in A$ . Recently, by using the probabilistic method, Alon answered two questions of Ljujić and Nathanson affirmatively by proving that, for $A = {n!}_{n \geq 1}$ or for $A = {n^{n}}_{n \geq 1}$ , there exists $n_{0}$ and an infinite set M of positive integers so that $0 < p (n, A, M) < n^{8 + o (1)}$ for all $n > n_{0}$ . In this note, by an explicit construction, as a corollary of our main result, it is proved that, for $A = {n!}_{n \geq 1}$ or for $A = {n^{n}}_{n \geq 1}$ , there exists an explicit infinite set M of positive integers so that $0 < p (n, A, M) \leq n^{2 + o (1)}$ for all $n \geq 1$ . Several open questions are posed for further research.

Soit N l'ensemble des entiers positifs ou nul. Pour $A, M \subset N ∖ {0}$ et $n \in N$ , notons $p (n, A, M)$ le nombre de représentations de n sous la forme $n = \sum_{a \in A} m_{a} a$ , avec $m_{a} \in M \cup {0}$ pour tout $a \in A$ . Récemment, utilisant une méthode probabiliste, Alon a répondu positivement à deux questions de Ljujić et Nathanson. Il a montré que, pour $A = {n!}_{n \geq 1}$ ou $A = {n^{n}}_{n \geq 1}$ , il existe $n_{0}$ et un ensemble infini M d'entiers positifs tel que $0 < p (n, A, M) < n^{8 + o (1)}$ pour tout $n > n_{0}$ . Dans cette Note, par une construction explicite et comme corollaire de notre résultat principal, nous montrons que, pour $A = {n!}_{n \geq 1}$ ou $A = {n^{n}}_{n \geq 1}$ , il existe un ensemble infini explicite M d'entiers positifs tel que $0 < p (n, A, M) < n^{2 + o (1)}$ pour tout $n \geq 1$ . Plusieurs questions ouvertes sont proposées pour de futures recherches.

Received: 2015-05-06
Accepted: 2016-01-07
Published online: 2016-02-09

DOI: 10.1016/j.crma.2016.01.007

Author's affiliations:

Li-Xia Dai ¹; Yong-Gao Chen ¹

¹ School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, PR China

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     author = {Li-Xia Dai and Yong-Gao Chen},
     title = {On two problems of {Ljuji\'c} and {Nathanson}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {235--238},
     publisher = {Elsevier},
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     number = {3},
     year = {2016},
     doi = {10.1016/j.crma.2016.01.007},
     language = {en},
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Li-Xia Dai; Yong-Gao Chen. On two problems of Ljujić and Nathanson. Comptes Rendus. Mathématique, Volume 354 (2016) no. 3, pp. 235-238. doi : 10.1016/j.crma.2016.01.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.01.007/

References
Cited by

[1] N. Alon Restricted integer partition functions, Integers, Volume 13 (2013), p. A16

[2] E.R. Canfield; H.S. Wilf On the growth of restricted integer partition functions, Dev. Math., Volume 23 (2012), pp. 39-46

[3] Z. Ljujić; M. Nathanson On a partition problem of Canfield and Wilf, Integers, Volume 12A (2012), p. A11

Cited by Sources:

^☆ This work was supported by the National Natural Science Foundation of China, Grant Nos. 11371195, 11271185, 11571174 and PAPD.

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