The discretised ABC sum-product conjecture

Let $\alpha,\beta,\gamma in (0,1)$ with $\beta \leq \alpha$. Let $A,B,C \subset [0,1]$ be $\delta$-separated sets with $|A| \leq \delta^{-\alpha}$, $|B|=\delta^{-\beta}$, and $|C|=\delta^{-\gamma}$. Assume the non-concentration conditions

$$|B \cap B(x,r)| lesssim r^{\beta}|B|,\qquad |C \cap B(x,r)| lesssim r^{\gamma}|C|$$

for all $x in \mathbb{R}$ and $r>0$. ABC sum-product conjecture. If $\gamma > \alpha-\beta$, then there exist $\epsilon=\epsilon(\alpha,\beta,\gamma)>0$ and $cin C$ such that

$$|A+cB|_{\delta} gtrsim_{\alpha,\beta,\gamma} \delta^{-\epsilon}|A|.$$

This is the discretised analogue of the ABC sum-product problem. The condition $\gamma>\alpha-\beta$ corresponds to the threshold $|B||C| gg |A|$ appearing in discrete variants, while the previously established result in the paper uses the stronger condition $\gamma>(\alpha-\beta)/(1-\beta)$; the conjectured sharp range remains open.