The Hausdorff-dimension neighbourhood estimate for missing digits measures

Let $n\geq2$ be an integer and let $M\subset\mathbb{R}^n$ be a manifold of finite type. Neighbourhood estimate conjecture. There exists a number $\sigma=\sigma(M)>0$ such that, for each missing digits measure $\lambda$ satisfying

$$\dim_{\mathrm{H}}\lambda>\sigma,$$

one has

$$\lambda(M^\delta)\ll\delta^{n-\dim M}.$$

The conjecture proposes replacing the Fourier $l^1$-dimension hypothesis by a Hausdorff-dimension hypothesis. The paper notes examples satisfying the conclusion, but only for large bases and specially structured digit sets; reducing those restrictions remains open.