[Ensembles sommes d’ensembles distincts d’entiers]
Let $A$ and $B$ be two non-empty finite subsets of integers and $n,k \in \mathbb{Z}^{+}$. The sum of dilates of sets $A$ and $B$ can be defined: $n \cdot A + k \cdot B = \bigl \lbrace na+kb: a \in A, \ b \in B\bigr \rbrace $. Under certain assumptions, in 2017, Mistri [Notes Number Theory Discrete Math. 23 (2017), no. 4] proved that $\vert {A + k \cdot B}\vert \ge 4\vert {A}\vert - 4$, where $A$ and $B$ are non-empty finite subsets of integers and $\vert {k}\vert \ge 3$. In this article, we consider the sumset $p \cdot A + k \cdot B$, with $k(>p)\in \mathbb{Z}^{+}$, where $p$ is an odd prime and $(p,k) = 1$, and prove that $\vert {p \cdot A + k \cdot B}\vert \ge 4 \vert {A}\vert - 4$, where $A$ and $B$ are non-empty finite subsets of integers satisfying some properties. We also rectify the proof of Kaur and Singh [Comptes Rendus. Mathématique 362 (2024)] and generalize this extended inverse result for arbitrary values of $k \in \mathbb{Z}^{+} (\ge 2)$.
Soient $A$ et $B$ deux sous-ensembles finis non vides de l’ensemble des entiers, et $n,k \in \mathbb{Z}^{+}$. La somme des dilatations des ensembles $A$ et $B$ peut être définie comme suit : $n \cdot A + k \cdot B = \bigl \lbrace na+kb: a \in A, \ b \in B\bigr \rbrace $. Sous certaines hypothèses, Mistri [Notes Number Theory Discrete Math. 23 (2017), no. 4] a prouvé en 2017 que $\vert {A + k \cdot B}\vert \ge 4\vert {A}\vert - 4$, où $A$ et $B$ sont des sous-ensembles finis non vides de l’ensemble des entiers et $\vert {k}\vert \ge 3$. Dans cet article, nous considérons l’ensemble somme $p \cdot A + k \cdot B$, avec $k(>p)\in \mathbb{Z}^{+}$, où $p$ est un nombre premier impair et $(p,k) = 1$, et nous démontrons que $\vert {p \cdot A + k \cdot B}\vert \ge 4 \vert {A}\vert - 4$, où $A$ et $B$ sont des sous-ensembles finis non vides de l’ensemble des entiers satisfaisant à certaines propriétés. Nous corrigeons également la démonstration de Kaur et Singh [Comptes Rendus. Mathématique 362 (2024)] et généralisons ce résultat inverse étendu pour toute valeur arbitraire de $k \in \mathbb{Z}^{+} (\ge 2)$.
Révisé le :
Accepté le :
Publié le :
Keywords: Sum of dilates, additive combinatorics, sumsets
Mots-clés : Somme des dilatés, combinatoire additive, ensembles-sommes
Gurpreet Kour 1 ; Sandeep Singh 1
CC-BY 4.0
Gurpreet Kour; Sandeep Singh. Sumset problem for distinct sets of integers. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 665-672. doi: 10.5802/crmath.857
@article{CRMATH_2026__364_G3_665_0,
author = {Gurpreet Kour and Sandeep Singh},
title = {Sumset problem for distinct sets of integers},
journal = {Comptes Rendus. Math\'ematique},
pages = {665--672},
year = {2026},
publisher = {Acad\'emie des sciences, Paris},
volume = {364},
doi = {10.5802/crmath.857},
language = {en},
}
[1] On the sum of dilations of a set, Acta Arith., Volume 164 (2014) no. 2, pp. 153-162 | DOI | MR
[2] On some direct and inverse results concerning sums of dilates, Acta Arith., Volume 188 (2019) no. 2, pp. 101-109 | DOI | MR
[3] Sums of dilates, Comb. Probab. Comput., Volume 17 (2008) no. 5, pp. 627-639 | DOI | MR
[4] Recherches sur les nombres, Œuvres complètes, Gauthier-Villars (1882–1974), pp. 39-63 Originally published in J. Éc. Polytech., Math. XVI (1813), t. IX, pp. 99–116
[5] On a sumset problem of dilates, Indian J. Pure Appl. Math., Volume 52 (2021) no. 4, pp. 1180-1185 | DOI | MR
[6] A theorem on the addition of residue classes: application to the number $\Gamma (k)$ in Waring’s problem, Proc. Indian Acad. Sci., Sect. A, Volume 2 (1935), pp. 242-243 | Zbl | DOI
[7] On sums of dilates, Comb. Probab. Comput., Volume 18 (2009) no. 6, pp. 871-880 | DOI | MR
[8] A sumset problem, J. Comb. Number Theory, Volume 2 (2010) no. 1, pp. 79-89 | MR
[9] On the addition of residue classes., J. Lond. Math. Soc., Volume 10 (1935), pp. 30-32 | DOI | MR
[10] On a sumset problem for integers, Electron. J. Comb., Volume 21 (2014) no. 1, 1.13, 25 pages | DOI | MR
[11] Direct and inverse problems in additive number theory and in non-abelian group theory, Eur. J. Comb., Volume 40 (2014), pp. 42-54 | DOI | MR
[12] A lower bound for the size of a Minkowski sum of dilates, Comb. Probab. Comput., Volume 20 (2011) no. 2, pp. 249-256 | DOI | MR
[13] On a sumset problem for dilated integer sets, Int. J. Number Theory, Volume 20 (2024) no. 7, pp. 1849-1862 | DOI | MR
[14] On direct and inverse problems related to some dilated sumsets, Comptes Rendus. Mathématique, Volume 362 (2024), pp. 99-105 | DOI | MR | Numdam
[15] On addition of two distinct sets of integers, Acta Arith., Volume 70 (1995) no. 1, pp. 85-91 | DOI | MR
[16] A lower bound for the size of a sum of dilates, J. Comb. Number Theory, Volume 5 (2013) no. 1, pp. 31-51 | MR
[17] Sums of dilates of two sets, Notes Number Theory Discrete Math., Volume 23 (2017) no. 4, pp. 34-41
[18] Additive number theory, Graduate Texts in Mathematics, 165, Springer, 1996 | MR
[19] On addition of two distinct sets of integers, Acta Arith., Volume 75 (1996) no. 2, pp. 191-194 | DOI | MR
Cité par Sources :
Commentaires - Politique