On the cardinality of general $h$-fold sumsets

Let $A=\{a_0,a_1,\dots ,a_{k-1}\}$ be a set of $k$ integers. For any integer $h\ge 1$ and any ordered $k$-tuple of positive integers $\mathbf{r}=(r_0,r_1,\dots ,r_{k-1})$, we define a general $h$-fold sumset, denoted by $h^{(\mathbf{r})}A$, which is the set of all sums of $h$ elements of $A$, where $a_i$ appearing in the sum can be repeated at most $r_i$ times for $i=0,1,\dots ,k-1$. In this paper, we give the best lower bound for $\mathrm{\mid }{h}^{(\mathbf{r})}A\mathrm{\mid }$ in terms of $\mathbf{r}$ and $h$ and determine the structure of the set $A$ when $\mathrm{\mid }{h}^{(\mathbf{r})}A\mathrm{\mid }$ is minimal. This generalizes results of Nathanson, and recent results of Mistri and Pandey and also solves a problem of Mistri and Pandey.