On a problem of Erdős and Graham

In this paper, we focus on an old problem of Erdős and Graham. Let $k\geq 3$ be an integer and $\mathcal{A}=(a_i)_{i=1}^\infty$ be a sequence of integers. Let $k\mathcal{A}$ be the set of all sums of $k$ elements of $\mathcal{A}$ with repetitions allowed. We show that if the difference sequence of $\mathcal{A}$ is block type, then there is sequence $\mathcal{B}$ such that $k\mathcal{A}\cap \mathcal{B}\neq\emptyset$.