 Polynomial skew-sumset expansion over a prime field

For positive $\alpha ,\beta$, suppose $p^{\alpha }\le |A|\le p^{1-\alpha }$ and $|B|\ge p^{\beta }$. 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 $|A+bA|\le K|A|$ to all these scalars with exponent depending only on the number of steps. Averaging collisions over the resulting large scalar set then forces $K^L\ge \frac{1}{4}\min (|A|,p/|A|)\ge p^{\alpha }/4$. 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 $1<|A|<p$.