arXiv subjects

Vivekanand Goswami

Publications and source records attributed to Vivekanand Goswami.

3 recordsLinked to original sources

arXiv2026

$S_h$-sets in abelian groups

For a positive integer $h$, a subset $A = \{a_1, \dots, a_k\}$ of an additive abelian group $G$ is called an $S_h$-set of size $k$ if all sums of $h$ distinct elements in $S$ are distinct. For fixed positive integers $h$ and $k$, let $v_h(k)$ denote the order of the smallest abelian group containing an $S_h$-set of size $k$. A lower bound for $v_2(k)$ is known. In this paper, we establish a lower bound for $v_3(k)$. Using our argument for $h=2$, we recover the known bound for $v_2(k)$.

math.NT↗

arXiv2026

Restricted set addition in finite abelian groups

Let $A$ be a nonempty subset of finite abelian group $G$ of order $n$. For an integer $h \geq 2$, the restricted $h$-fold sumset $h^\wedge A$ is the set of all sums of $h$ distinct elements of $A$. It is known that if $G$ is a group of order $n$ and $A$ is a subset of $G$ such that $|A|$ is close to $\frac{n}{2}$, then $h^{\wedge}A = G$ under some conditions on $h$ and $n$. The constant $\frac{1}{2}$ is optimal for groups of even order but not for groups of odd order. For an integer $h \geq 4$, let $α_h$ be the unique positive root of the polynomial $3^{h - 2} x^{h - 1} + x - 1$. In this paper, we show that for any $α> α_h$, there exists a positive integer $M_h(α)$, which is determined precisely, such that for all $n > M_h(α)$ with $n$ odd, if $A$ is a subset of a finite abelian group $G$ of order $n$ and if $|A| \geq αn$, then $h^{\wedge} A = G$. Moreover, $α_h > α_{h + 1}$ for $h \geq 4$ and $α_h$ approaches $\frac{1}{3}$ as $h$ increases, and the constant $\frac{1}{3}$ is optimal when the smallest prime dividing $n$ is $3$. This result extends a theorem of Tang and Wei on $4^{\wedge}A$ in the cyclic group $\mathbb{Z}_n$ to $h^{\wedge}A$ for every $h \geq 4$, and to arbitrary finite abelian groups.

math.NT↗

arXiv2025

The maximum size of sumsets in finite cyclic groups

Let $A$ be a nonempty finite subset of an additive abelian group $G$. Given a nonnegative integer $h$, the $h$-fold sumset $hA$ is the set of all sums of $h$ elements of $A$, and the restricted $h$-fold sumset $h^\wedge A$ is the set of all sums of $h$ distinct elements of $A$. The union of restricted sumsets $s^\wedge A$, where $s=0, 1, \ldots, h$, is denoted by $[0, h]^\wedge A$. For fixed positive integers $m$ and $h$, the maximum size of the sumset $hA$ of a set $A \subseteq G$ with $m$ elements is denoted by $ν(G, m, h)$. In other words, $ν(G, m, h) = \max\{|hA| : A \subseteq G, |A|= m\}$. Analogous quantities can be defined for the sumsets $h^\wedge A$ and $[0, h]^\wedge A$. Optimal upper bounds are known for these quantities. If $G$ is a finite cyclic group of order $n$, then each of these quantities agrees with the optimal upper bound, except in many cases. Bajnok posed the problem of determining all positive integers $n$, $m$, and $h$ for which the value of the function $f(n, m, h)$ is strictly less than the optimal upper bound. He posed similar problems for quantities related to the sumsets $h^\wedge A$ and $[0, h]^\wedge A$. We prove that, for any positive integer $h$, there are infinitely many positive integers $m$ and $n$ such that $ν(\mathbb{Z}_n, m, h)$ is strictly less than the optimal upper bound. We also prove similar results for quantities related to the sumsets $h^\wedge A$ and $[0, h]^\wedge A$ also. These results provide the partial solutions to the problems posed by Bajnok.

math.NT↗