On the cardinality of subsequence sums II

Jiang, Xing-Wang; Li, Ya-Li

Comptes Rendus. Mathématique, Tome 362 (2024) no. G11, pp. 1279-1285

Let $A=({\underbrace{a_1,...,a_1}}_{r_1},{\underbrace{a_2,...,a_2}}_{r_2},...,{\underbrace{a_k,...,a_k}}_{r_k})$ be a finite sequence of integers with $a_1<a_2<\cdots <a_k$ and $r_i\ge 1(1\le i\le k)$. The sum of all terms of a subsequence $B$ of $A$ is called a subsequence sum of $A$ and we denote it by $\sigma \left(B\right)$. For $0\le \alpha \le {\sum }_{i=1}^k{r}_i$, let ${\Sigma }_{\alpha }\left(A\right)=\left\{\sigma \left(B\right)\right|B\text{is}\text{a}\text{subsequence}\text{of}A\text{of}\text{length}\ge \alpha \}$. In this paper, we completely settle the two problems posed by Bhanja and Pandey about finding the optimal lower bound of $|{\Sigma }_{\alpha }\left(A\right)|$ and determining the structure of the sequence $A$ for which the lower bound of $|{\Sigma }_{\alpha }\left(A\right)|$ is optimal.

Soit $A=({\underbrace{a_1,...,a_1}}_{r_1},{\underbrace{a_2,...,a_2}}_{r_2},...,{\underbrace{a_k,...,a_k}}_{r_k})$) une suite finie d’entiers avec $a_1<a_2<\cdots <a_k$ et $r_i\ge 1(1\le i\le k)$. La somme de tous les termes d’une sous-suite $B$ de $A$ est appelée somme de sous-suite de $A$ et nous la désignons par $\sigma \left(B\right)$. Pour $0\le \alpha \le {\sum }_{i=1}^k{r}_i$, soit ${\Sigma }_{\alpha }\left(A\right)=\left\{\sigma \left(B\right)\right|B\text{une}\text{sous-suite}\text{de}A\text{de}\text{longueur}\ge \alpha \}$. Dans cet article, nous résolvons complètement les deux problèmes posés par Bhanja et Pandey concernant la recherche de la borne inférieure de $|{\Sigma }_{\alpha }\left(A\right)|$ et la détermination de la structure de la suite $A$ pour laquelle la borne inférieure de $|{\Sigma }_{\alpha }\left(A\right)|$ est atteinte.

Classification : 11P70, 11B25
Keywords: Subsequence sums, direct problem, inverse problem
Mots-clés : Sommes de sous-suites, problème direct, problème inverse

[1] Bhanja, Jagannath A note on sumsets and restricted sumsets, J. Integer Seq., Volume 24 (2021) no. 4, 21.4.2, 9 pages | MR | Zbl

[2] Bhanja, Jagannath; Pandey, Ram Krishna Inverse problems for certain subsequence sums in integers, Discrete Math., Volume 343 (2020) no. 12, 112148, 11 pages | DOI | MR | Zbl

[3] Bhanja, Jagannath; Pandey, Ram Krishna On the minimum size of subset and subsequence sums in integers, C. R. Math. Acad. Sci. Paris, Volume 360 (2022), pp. 1099-1111 | DOI | MR | Zbl | Numdam

[4] Dwivedi, Himanshu Kumar; Mistri, Raj Kumar Direct and inverse problems for subset sums with certain restrictions, Integers, Volume 22 (2022), A112, 13 pages | MR | Zbl

[5] Jiang, Xing-Wang; Li, Ya-Li On the cardinality of subsequence sums, Int. J. Number Theory, Volume 14 (2018) no. 3, pp. 661-668 | DOI | MR | Zbl

[6] Mistri, Raj Kumar; Pandey, Ram Krishna A generalization of sumsets of set of integers, J. Number Theory, Volume 143 (2014), pp. 334-356 | DOI | MR | Zbl

[7] Mistri, Raj Kumar; Pandey, Ram Krishna; Prakash, Om Subsequence sums: direct and inverse problems, J. Number Theory, Volume 148 (2015), pp. 235-256 | DOI | MR | Zbl

[8] Nathanson, Melvyn B. Inverse theorems for subset sums, Trans. Am. Math. Soc., Volume 347 (1995) no. 4, pp. 1409-1418 | DOI | MR | Zbl

[9] Nathanson, Melvyn B. Additive number theory. Inverse problems and the geometry of sumsets, Graduate Texts in Mathematics, 165, Springer, 1996, xiv+293 pages | DOI | MR | Zbl

[10] Vu, Van H. Some new results on subset sums, J. Number Theory, Volume 124 (2007) no. 1, pp. 229-233 | DOI | MR | Zbl

[11] Yang, Quan-Hui; Chen, Yong-Gao On the cardinality of general $h$-fold sumsets, Eur. J. Comb., Volume 47 (2015), pp. 103-114 | DOI | MR | Zbl