Is it possible to construct a $B$ $\subseteq$ $Z_p(=Z/pZ)$ of cardinal $cp^{\frac{1}{3}}$, for some constant $c$, such that there exists an arithmetic progression of size $c_1p^{\frac{2}{3}}$, for some constant $c_1$, inside $B+αB$ for any $\alpha\in Z_p$?
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$\begingroup$ What is wrong with taking $B$ to be an arithmetic progression of size $cp^ {1/3}$? $\endgroup$– js21Feb 13, 2018 at 15:28
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$\begingroup$ @js21 $B+B$ will then have size around $2cp^{1/3}$. $\endgroup$– WojowuFeb 13, 2018 at 17:41
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$\begingroup$ Oh, I misread the exponent $2/3$. $\endgroup$– js21Feb 15, 2018 at 10:25
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