Let $A, B \subset \mathbb R$ be IID random subset. Suppose that there exists $d \in (1/2, 1]$ such that the Hausdorff dimension of $A$ is equal to $d$ almost surely. Is it true that $\mathbf P\lbrack A \cap B \neq \emptyset\rbrack > 0$?

(Note that zero sets of two independent Brownian motions almost surely do not intersect. Thus we have to require at least $d > 1/2$.)

Let $A, B \subset \mathbb R$ be IID random closed subset. Suppose that there exists $d \in (1/2, 1]$ such that the Hausdorff dimension of $A$ is equal to $d$ almost surely. Is it true that $\mathbf P\lbrack A \cap B \neq \emptyset\rbrack > 0$?

(Note that zero sets of two independent Brownian motions almost surely do not intersect. Thus we have to require at least $d > 1/2$.)

Let $A, B \subset \mathbb R$ be IID random subset. Suppose that there exists $d \in (1/2, 1]$ such that the Hausdorff dimension of $A$ is equal to $d$ almost surely. Is it true that $\mathbf P\lbrack A \cap B \neq \emptyset\rbrack > 0$?

Let $A, B \subset \mathbb R$ be IID random subset. Suppose that there exists $d \in (1/2, 1]$ such that the Hausdorff dimension of $A$ is equal to $d$ almost surely. Is it true that $\mathbf P\lbrack A \cap B \neq \emptyset\rbrack > 0$?

(Note that zero sets of two independent Brownian motions almost surely do not intersect. Thus we have to require at least $d > 1/2$.)