We define the *lower density* of a set $A\subseteq \mathbb{N}$ by
$$
\operatorname{ld}(A) \ := \ \liminf_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}.
$$
For $A,B\subseteq \mathbb{N}$, we set
$$
A + B \ := \ \{a+b: a\in A, b\in B\},
$$
and similarly
$$
A \cdot B \ := \ \{a\cdot b: a\in A, b\in B\}.
$$
Further let ${\cal P}_0(\mathbb{N}) := \{A\in{\cal P}(\mathbb{N}): \operatorname{ld}(A) = 0\}$ be the set of all sets of positive integers
whose lower density is $0$.

**Question:** What are

- $\text{sup}\{\text{ld}(A+B): A,B\in {\cal P}_0(\mathbb{N})\}$, and
- $\text{sup}\{\text{ld}(A\cdot B): A,B\in {\cal P}_0(\mathbb{N})\}$?