Furstenberg's slicing conjecture

Let $X,Y\subset [0,1)$ be closed and invariant under $T_a,T_b$ respectively, where $T_m(x)=mx\bmod 1$ is multiplication by $m$ on the circle. Assume that $\log a/\log b$ is irrational. Furstenberg's slicing conjecture. For every line $\ell$ that is neither vertical nor horizontal,

$$\operatorname{dim_H}((X\times Y)\cap \ell)\leq \max(\operatorname{dim_H}(X)+\operatorname{dim_H}(Y)-1,0).$$

This conjecture concerns the dimension of slices of products of dynamically defined sets and was resolved independently by Pablo Shmerkin and M. Wu.