Geodesic
Sum and difference sets containing integer powers
Czechoslovak Mathematical Journal, Tome 62 (2012) no. 3, pp. 787-793 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.1007/s10587-012-0045-2
Classification : 11B13, 11B30
Keywords: sum and difference set; integer power 
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Yang, Quan-Hui; Wu, Jian-Dong. Sum and difference sets containing integer powers. Czechoslovak Mathematical Journal, Tome 62 (2012) no. 3, pp. 787-793. doi: 10.1007/s10587-012-0045-2

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