Projection theorems with countably many exceptions and applications to the exact overlaps conjecture
Meng Wu
TL;DR
The paper advances projection theory for fractal measures by establishing that, under a $\beta$-transversality regime, the set of parameters or directions yielding a dimension drop is at most countable. It leverages CP-distributions, Hochman’s entropy inverse theorem, and additive combinatorics to prove dimension stability for projections of self-similar measures and for CP-distributions, with broad applications to packing and Assouad dimensions. Key contributions include a precise countability result for exceptional projections, a sharp projection theorem for measures with uniform entropy dimension, and a corresponding sharp result for Assouad-dimension projections, along with a detailed organizational framework and proofs. The results have implications for the exact overlaps conjecture and provide robust tools for analyzing projections in fractal geometry and dynamical systems, with potential reach to a wide class of self-similar and regular measures.
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 $α$, 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.