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Let 
$S=\{s_{1}, s_{2}, \ldots \}$ be an unbounded sequence of positive integers with 
$s_{n+1}/s_{n}$ approaching 
$\alpha $ as 
$n\rightarrow \infty $ and let 
$\beta>\max (\alpha , 2)$. We show that for all sufficiently large positive integers l, if 
$A\subset [0, l]$ with 
$l\in A$, 
$\gcd A=1$ and 
$|A|\geq (2-{k}/{\lambda \beta })l/(\lambda +1)$, where 
$\lambda =\lceil {k}/{\beta }\rceil $, then 
$kA\cap S\neq \emptyset $ for 
$2<\beta \leq 3$ and 
$k\geq {2\beta }/{(\beta -2)}$ or for 
$\beta>3$ and 
$k\geq 3$.
