A discretised projection theorem in the plane

The main result of this paper is that for any $1/2 \leq s < 2 - \sqrt{2} \approx 0.5858$, there is a number $\sigma = \sigma(s) < s$ with the following property. Let $\delta > 0$ be small, assume that $A \subset [0,1]$ is a $(\delta,1/2)$-set, and that $E \subset [0,1]$ contains $\gtrsim \delta^{-\sigma}$ roughly $\delta^{s}$-separated points. Then there exists a number $t \in E$ such that $A + tA$ contains $\gtrsim \delta^{-s}$ $\delta$-separated points. For $\sigma = s$, this is essentially a consequence of Kaufman's well-known bound for exceptional sets of projections. Our proof consists of a structural observation concerning sets, for which Kaufman's bound is near-optimal, combined with (an adaptation of) Solymosi's argument for his "$4/3$" sum-product theorem.