The small-set function restricted-sumset conjecture

Let $p$ be a prime number, and let $A,B\subseteq\mathbb{F}_p$ satisfy $|B|\leqslant |A|$. Let $\mathcal{R}\colon B\to A$ be an arbitrary function, and define

$$A+_{\mathcal{R}}B=\{a+b:a\in A,\ b\in B,\ a\ne\mathcal{R}(b)\}.$$

Small-set restricted-sumset conjecture. If

$$|A|+2|B|\leqslant p,$$

then

$$|A+_{\mathcal{R}}B|\geqslant |A|+|B|-3.$$

This extends the matching case to arbitrary functions, allowing repeated forbidden partners. The paper identifies this regime as open; its main results prove related bounds under stronger quantitative hypotheses.