A conjecture on arithmetic sums of Ahlfors-regular sets

Let $\kappa>0$, and let $A,B,E\subset[0,1]$ be compact sets. Write $\dim_{\mathrm{H}}$ for Hausdorff dimension, and let denote the Bourgain estimate discussed in the paper.

General arithmetic-sum conjecture. If

$$\dim_{\mathrm{H}} B\geq\kappa,\qquad \dim_{\mathrm{H}} B+\dim_{\mathrm{H}} E\geq\dim_{\mathrm{H}} A+\kappa,\qquad \dim_{\mathrm{H}} A\leq 1-\kappa,$$

then holds for some $\theta\in E$, and for some $\epsilon>0$ depending only on $\kappa$.

The conjecture extends Bourgain's result from the case of equal parameters to the unequal-parameter setting, motivated by evidence over finite fields. The precise estimate referenced as is not included in the supplied context.