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 } ≤ α − β 1 − β + κ . \begin{equation*} \dim _{\mathrm {H}} \{c \in \mathbb {R}: \dim _{\mathrm {H}} (A + cB) \leq \alpha + \eta \} \leq \tfrac {\alpha - \beta }{1 - \beta } + \kappa . \end{equation*} This extends a result of Bourgain from 2010, which contained the case α = β \alpha = \beta . The paper also contains a δ \delta -discretised, and somewhat stronger, version of the estimate above, and new information on the size of long sums of the form a 1 B + … + a n B a_{1}B + \ldots + a_{n}B .