Sum and difference sets containing integer powers

summary:Let $n > m \geq 2$ be positive integers and $n=(m+1) \ell +r$, where $0 \leq r \leq m.$ Let $C$ be a subset of $\{0,1,\cdots ,n\}$. We prove that if $$ |C|>\begin {cases} \lfloor n/2 \rfloor +1 &\text {if $m$ is odd}, \\ m \ell /2 +\delta &\text {if $m$ is even},\\ \end {cases} $$ where $\lfloor x \rfloor $ denotes the largest integer less than or equal to $x$ and $\delta $ denotes the cardinality of even numbers in the interval $[0,\min \{r,m-2\}]$, then $C-C$ contains a power of $m$. We also show that these lower bounds are best possible