On the discretised 𝐴𝐵𝐶 sum-product problem

Let 0 > β ≤ α > 1 0 > \beta \leq \alpha > 1 and κ > 0 \kappa > 0 . I prove that there exists η > 0 \eta > 0 such that the following holds for every pair of Borel sets A , B ⊂ R A,B \subset \mathbb {R} with dim H ⁡ A = α \dim _{\mathrm {H}} A = \alpha and dim H ⁡ B = β \dim _{\mathrm {H}} B = \beta : dim H ⁡ { c ∈ R : dim H ⁡ ( A + c B ) ≤ α + η } ≤