Sum and difference sets containing integer powers

Yang, Quan-Hui ; Wu, Jian-Dong

Czechoslovak Mathematical Journal, Tome 62 (2012) no. 3, pp. 787-793.

Résumé

Let $n>m\ge 2$ be positive integers and $n=(m+1)\ell +r$, where $0\le r\le m.$ 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.

Détail

DOI : 10.1007/s10587-012-0045-2

Classification : 11B13, 11B30
Mots-clés : sum and difference set; integer power