Projection theorems with countably many exceptions and applications to the exact overlaps conjecture
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912635337834496 |
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| author | Wu, Meng |
| author_facet | Wu, Meng |
| contents | 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 $α$, the packing dimension of its orthogonal projection is at least $\min\{1, α\}$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_21923 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Projection theorems with countably many exceptions and applications to the exact overlaps conjecture Wu, Meng Dynamical Systems Classical Analysis and ODEs Metric Geometry 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 $α$, the packing dimension of its orthogonal projection is at least $\min\{1, α\}$ 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. |
| title | Projection theorems with countably many exceptions and applications to the exact overlaps conjecture |
| topic | Dynamical Systems Classical Analysis and ODEs Metric Geometry |
| url | https://arxiv.org/abs/2503.21923 |