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For a nonempty finite set $A$ of integers, let $S(A) = \left\{ \sum_{b\in B} b: \emptyset \not= B\subseteq A\right\}$ be the set of all nonempty subset sums of $A$. In 1995, Nathanson determined the minimum cardinality of $S(A)$ in terms of $|A|$ and
Externí odkaz:
http://arxiv.org/abs/2401.08208
Autor:
Mohan, Pandey, Ram Krishna
Let $h \geq 2$, $k \geq 5$ be integers and $A$ be a nonempty finite set of $k$ integers. Very recently, Tang and Xing studied extended inverse theorems for $hk-h+1 < \left|hA\right| \leq hk+2h-3$. In this paper, we extend the work of Tang and Xing an
Externí odkaz:
http://arxiv.org/abs/2401.04699
Autor:
Hegyvári, Norbert
Erd\H os introduced the quantity $S=T\sum^T_{i=1}X_i$, where $X_1,\dots, X_T$ are arithmetic progressions, and cover the square numbers up to $N$. He conjectured that $S$ is close to $N$, i.e. the square numbers cannot be covered "economically" by ar
Externí odkaz:
http://arxiv.org/abs/2302.00408
Autor:
Bhanja, Jagannath, Pandey, Ram Krishna
Publikováno v:
Comptes Rendus Mathematique, Volume 360 (2022), pp. 1099-1111
Let $\mathcal{A}$ be a sequence of $rk$ terms which is made up of $k$ distinct integers each appearing exactly $r$ times in $\mathcal{A}$. The sum of all terms of a subsequence of $\mathcal{A}$ is called a subsequence sum of $\mathcal{A}$. For a nonn
Externí odkaz:
http://arxiv.org/abs/2108.07042
Autor:
Bhanja, Jagannath
Publikováno v:
Journal of Integer Sequences, 24 (2021), Article 21.4.2
In this note we find the optimal lower bound for the size of the sumsets $HA$ and $H\,\hat{}A$ over finite sets $H, A$ of nonnegative integers, where $HA = \bigcup_{h\in H} hA$ and $H\,\hat{}A = \bigcup_{h\in H} h\,\hat{}A$. We also find the underlyi
Externí odkaz:
http://arxiv.org/abs/2106.04091
Autor:
Bhanja, Jagannath, Pandey, Ram Krishna
Let $A$ be a nonempty finite set of $k$ integers. Given a subset $B$ of $A$, the sum of all elements of $B$, denoted by $s(B)$, is called the subset sum of $B$. For a nonnegative integer $\alpha$ ($\leq k$), let \[\Sigma_{\alpha} (A):=\{s(B): B \subs
Externí odkaz:
http://arxiv.org/abs/1909.00194
Let $G$ be an additive abelian group. Let $A=\{a_{0}, a_{1},\ldots, a_{k-1}\}$ be a nonempty finite subset of $G$. For a positive integer $h$ satisfying $1\leq h\leq k$, we let \[h\hat{}_{\underline{+}}A:=\{\Sigma_{i=0}^{k-1}\lambda_{i} a_{i}: (\lamb
Externí odkaz:
http://arxiv.org/abs/1908.00081
Autor:
Rué, Juanjo, Spiegel, Christoph
We prove that for pairwise co-prime numbers $k_1,\dots,k_d \geq 2$ there does not exist any infinite set of positive integers $A$ such that the representation function $r_A (n) = \{ (a_1, \dots, a_d) \in A^d : k_1 a_1 + \dots + k_d a_d = n \}$ become
Externí odkaz:
http://arxiv.org/abs/1802.07597
We give asymptotic sharp estimates for the cardinality of a set of residue classes with the property that the representation function is bounded by a prescribed number. We then use this to obtain an analogous result for sets of integers, answering an
Externí odkaz:
http://arxiv.org/abs/0909.5024
Autor:
Jagannath Bhanja
Publikováno v:
Scopus-Elsevier
In this note we find the optimal lower bound for the size of the sumsets $HA$ and $H\,\hat{}A$ over finite sets $H, A$ of nonnegative integers, where $HA = \bigcup_{h\in H} hA$ and $H\,\hat{}A = \bigcup_{h\in H} h\,\hat{}A$. We also find the underlyi
Externí odkaz:
https://explore.openaire.eu/search/publication?articleId=doi_dedup___::12ba6be28f875e19271b0eb5dc0902ae
http://arxiv.org/abs/2106.04091
http://arxiv.org/abs/2106.04091