Projection theorems with countably many exceptions and applications to the exact overlaps conjecture

We establish several optimal estimates for exceptional parameters in the projection of fractal measures: (1) For a parametric family of self-similar measures satisfying a transversality condition, the set of parameters leading to a dimension drop is at most countable. (2) For any ergodic CP-distribution $Q$ on ${\mathbb{R}}^2$, the Hausdorff dimension of its orthogonal projection is $\min \{1,\dim Q\}$ in all but at most countably many directions. Applications of our projection results include: (i) For any planar Borel probability measure with uniform entropy dimension $\alpha$, the packing dimension of its orthogonal projection is at least $\min \{1,\alpha \}$ in all but at most countably many directions. (ii) For any planar set $F$, the Assouad dimension of its orthogonal projection is at least $\min \{1,{\dim }_{\mathrm{A}}F\}$ in all but at most countably many directions.