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arXiv · 2503.21923

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

Meng Wu
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Abstract

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_{\rm A} F\} $ in all but at most countably many directions.

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Meng Wu. 2025-03-27. Projection theorems with countably many exceptions and applications to the exact overlaps conjecture. https://arxiv.org/abs/2503.21923

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