Sum and difference sets containing integer powers

@article{Yang2012,
abstract = {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 $\lbrace 0,1,\cdots ,n\rbrace $. We prove that if \[ |C|>\{\left\lbrace \begin\{array\}\{ll\} \lfloor n/2 \rfloor +1 &\text\{if $m$ is odd\}, \\ m \ell /2 +\delta &\text\{if $m$ is even\},\\ \end\{array\}\right.\} \] 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 $[0,\min \lbrace r,m-2\rbrace ]$, then $C-C$ contains a power of $m$. We also show that these lower bounds are best possible.},
author = {Yang, Quan-Hui, Wu, Jian-Dong},
journal = {Czechoslovak Mathematical Journal},
keywords = {sum and difference set; integer power; sumset; difference set; integer power},
language = {eng},
number = {3},
pages = {787-793},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Sum and difference sets containing integer powers},
url = {http://eudml.org/doc/246808},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Yang, Quan-Hui
AU - Wu, Jian-Dong
TI - Sum and difference sets containing integer powers
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 3
SP - 787
EP - 793
AB - 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 $\lbrace 0,1,\cdots ,n\rbrace $. We prove that if \[ |C|>{\left\lbrace \begin{array}{ll} \lfloor n/2 \rfloor +1 &\text{if $m$ is odd}, \\ m \ell /2 +\delta &\text{if $m$ is even},\\ \end{array}\right.} \] 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 $[0,\min \lbrace r,m-2\rbrace ]$, then $C-C$ contains a power of $m$. We also show that these lower bounds are best possible.
LA - eng
KW - sum and difference set; integer power; sumset; difference set; integer power
UR - http://eudml.org/doc/246808
ER -