The dimension-conserving projections conjecture for near-regular measures

Let $u$ be a near-$d$-regular measure on $\mathbb{R}^{2}$, let $0\leq s<d$, and let $\mathsigma$ be a Borel probability measure on $S^{1}$. Suppose that for $\mathsigma$-almost every $e\in S^{1}$, the projection $\mathpi_{e}( u)$ is exact dimensional with dimension $s$, and the projections $\mathpi_{e}$ are dimension conserving relative to $u$. Dimension-conserving projections conjecture. Under these assumptions, $\mathop{\dim}_{\mathrm{H}}\mathsigma=0$. This is the key contradiction statement needed in the projection argument: if the relevant discretised projection and fibre properties hold for a near-$d$-regular tangent measure, then the exceptional set of directions cannot support positive Hausdorff dimension. The paper presents the claim as the rough formulation of the main combinatorial argument, while the precise hypotheses are given by the discretised conditions referred to as (K1)–(K2).