Sumsets with a minimum number of distinct terms

For a non-empty $k$-element set $A$ of an additive abelian group $G$ and a positive integer $r \leq k$, we consider the set of elements of $G$ that can be written as a sum of $h$ elements of $A$ with at least $r$ distinct elements. We denote this set as $h^{(\geq r)}A$ for integers $h \geq r$. The set $h^{(\geq r)}A$ generalizes the classical sumsets $hA$ and $h\hat{}A$ for $r=1$ and $r=h$, respectively. Thus, we call the set $h^{(\geq r)}A$ the generalized sumset of $A$. By writing the sumset $h^{(\geq r)}A$ in terms of the sumsets $hA$ and $h\hat{}A$, we obtain the sharp lower bound on the size of $h^{(\geq r)}A$ over the groups $\mathbb{Z}$ and $\mathbb{Z}_p$, where $p$ is a prime number. We also characterize the set $A$ for which the lower bound on the size of $h^{(\geq r)}A$ is tight in these groups. Further, using some elementary arguments, we prove an upper bound for the minimum size of $h^{(\geq r)}A$ over the group $\mathbb{Z}_m$ for any integer $m \geq 2$.Comment: 16 page

A note on distinct differences in tt-intersecting families

For a family $\mathcal{F}$ of subsets of $\{1,2,\ldots,n\}$, let $\mathcal{D}(\mathcal{F}) = \{F\setminus G: F, G \in \mathcal{F}\}$ be the collection of all (setwise) differences of $\mathcal{F}$. The family $\mathcal{F}$ is called a $t$-intersecting family, if for some positive integer $t$ and any two members $F, G \in \mathcal{F}$ we have $|F\cap G| \geq t$. The family $\mathcal{F}$ is simply called intersecting if $t=1$. Recently, Frankl proved an upper bound on the size of $\mathcal{D}(\mathcal{F})$ for the intersecting families $\mathcal{F}$. In this note we extend the result of Frankl to $t$-intersecting families

An improved threshold for the number of distinct intersections of intersecting families

A family $\mathcal{F}$ of subsets of $\{1,2,\ldots,n\}$ is called a $t$-intersecting family if $|F\cap G| \geq t$ for any two members $F, G \in \mathcal{F}$ and for some positive integer $t$. If $t=1$, then we call the family $\mathcal{F}$ to be intersecting. Define the set $\mathcal{I}(\mathcal{F}) = \{F\cap G: F, G \in \mathcal{F} \text{ and } F \neq G\}$ to be the collection of all distinct intersections of $\mathcal{F}$. Frankl et al. proved an upper bound for the size of $\mathcal{I}(\mathcal{F})$ of intersecting families $\mathcal{F}$ of $k$-subsets of $\{1,2,\ldots,n\}$. Their theorem holds for integers $n \geq 50 k^2$. In this article, we prove an upper bound for the size of $\mathcal{I}(\mathcal{F})$ of $t$-intersecting families $\mathcal{F}$, provided that $n$ exceeds a certain number $f(k,t)$. Along the way we also improve the threshold $k^2$ to $k^{3/2+o(1)}$ for the intersecting families.Comment: Some errors in the previous draft have been correcte

Direct and inverse problems for restricted signed sumsets in integers

Let $A=\{a_0, a_1,\ldots, a_{k-1}\}$ be a nonempty finite subset of an additive abelian group $G$. For a positive integer $h$ $(\leq k)$, we let $h^{\wedge}_{\pm}A = \{\Sigma_{i=0}^{k-1} \lambda_{i} a_{i}: \lambda_{i} \in \{-1,0,1\} \text{ for } i=0, 1, \ldots, k-1,~~\Sigma_{i=0}^{k-1} |\lambda_{i}|=h\},$ be the $h$-fold restricted signed sumset of $A$. The direct problem for the restricted signed sumset is to find the minimum number of elements in $h^{\wedge}_{\pm}A$ in terms of $\lvert A\rvert$, where $\lvert A\rvert$ is the cardinality of $A$. The {\it inverse problem} for the restricted signed sumset is to determine the structure of the finite set $A$ for which the minimum value of $|h^{\wedge}_{\pm}A|$ is achieved. In this article, we solve some cases of both direct and inverse problems for $h^{\wedge}_{\pm}A$ in the group of integers. In this connection, we also mention some conjectures in the remaining cases

Direct and inverse problems for restricted signed sumsets in integers

Let $A=\{a_0, a_1,\ldots, a_{k-1}\}$ be a nonempty finite subset of an additive abelian group $G$. For a positive integer $h$ $(\leq k)$, we let $h^{\wedge}_{\pm}A = \{\Sigma_{i=0}^{k-1} \lambda_{i} a_{i}: \lambda_{i} \in \{-1,0,1\} \text{ for } i=0, 1, \ldots, k-1,~~\Sigma_{i=0}^{k-1} |\lambda_{i}|=h\},$ be the $h$-fold restricted signed sumset of $A$. The direct problem for the restricted signed sumset is to find the minimum number of elements in $h^{\wedge}_{\pm}A$ in terms of $\lvert A\rvert$, where $\lvert A\rvert$ is the cardinality of $A$. The {\it inverse problem} for the restricted signed sumset is to determine the structure of the finite set $A$ for which the minimum value of $|h^{\wedge}_{\pm}A|$ is achieved. In this article, we solve some cases of both direct and inverse problems for $h^{\wedge}_{\pm}A$ in the group of integers. In this connection, we also mention some conjectures in the remaining cases

On the minimum size of subset and subsequence sums in integers

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 nonnegative integer $\alpha \le rk$, let $\Sigma _{\alpha } (\mathcal{A})$ be the set of all subsequence sums of $\mathcal{A}$ that correspond to the subsequences of length $\alpha$ or more. When $r=1$, we call the subsequence sums as subset sums and we write $\Sigma _{\alpha } (A)$ for $\Sigma _{\alpha } (\mathcal{A})$. In this article, using some simple combinatorial arguments, we establish optimal lower bounds for the size of $\Sigma _{\alpha } (A)$ and $\Sigma _{\alpha } (\mathcal{A})$. As special cases, we also obtain some already known results in this study