 Restricted sumset bound for unequal subsets of a prime field

Suppose $|A|>|B|>0$ in ${\mathbb{F}}_p$. Put $a=|A|$ and $b^{\prime }=\min (|B|,p+2-a)$, and choose $B^{\prime }\subseteq B$ of size $b^{\prime }$. Let $r=a+b^{\prime }-2$. If the restricted sumset fits in a set $D$ of size $r-1$, then $(x-y){\prod }_{d\in D}(x+y-d)$ vanishes on $A\times B^{\prime }$. Its coefficient of $x^{a-1}{y}^{b^{\prime }-1}$ isAll factorial arguments are below $p$ and $0<a-b^{\prime }<p$. The Combinatorial Nullstellensatz gives a contradiction. Since $r=\min (p,|A|+|B|-2)$, this includes the saturated case as well as the small-set case.