For $A\subseteq \mathbb{N}$ weWe define the lower density by $$\operatorname{ld}(A)=\liminf_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}.$$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 $$ A + B \ := \ \{a+b: a\in A, b\in B\}, $$ and similarly $$ A \cdot B \ := \ \{a\cdot b: a\in A, b\in B\}. $$ Further let $A\cdot B = \{a\cdot b: a\in A, b\in B\}$. 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 ${\cal P}_0(\mathbb{N}) = \{A\in{\cal P}(\mathbb{N}): \operatorname{ld}(A) = 0\}$$0$.
Question:. What are
- $\text{sup}\{\text{ld}(A+B): A,B\in {\cal P}_0(\mathbb{N})\}$, and
- $\text{sup}\{\text{ld}(A\cdot B): A,B\in {\cal P}_0(\mathbb{N})\}$?
