= Polynomial skew-sumset expansion over a prime field
{title2=$\max_{b\in B}|A+bA|\geq|A|^{1+c_{\alpha,\beta}}$}
For positive $\alpha,\beta$, suppose $p^\alpha\leq|A|\leq p^{1-\alpha}$ and $|B|\geq p^\beta$. Repeatedly replace a <scalar> set $S$ by $3SS-3SS$. <Quadratic growth of a sixfold product difference set> reaches size at least $p/2$ in a number of steps depending only on $\beta$. The <closure bounds for small skew-sumset scalars> transfer a putative bound $|A+bA|\leq K|A|$ to all these <scalars> with exponent depending only on the number of steps. Averaging collisions over the resulting large <scalar> set then forces $K^L\geq\frac14\min(|A|,p/|A|)\geq p^\alpha/4$. This supplies a positive exponent depending only on $\alpha,\beta$. The finitely many smaller primes are covered by the strict inequality $|A+bA|>|A|$ for every nonzero $b$ when $1<|A|<p$.
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