How large can the upper density of a set avoiding $n\mapsto n^2$ be?
问题内容
Let $A\subset\mathbb{N}$ have positive upper density. Must there exist infinitely many $n$ such that both $n$ and $n^2$ belong to $A$?
If not, determine the largest possible upper density of a set $A$ for which $$ A\cap\{n^2:n\in A\} $$ is finite.
Can anyone please help me with this?
回答 (1)
The set $A$ of nonsquares has density $1$, and it is disjoint from $\{n^2:n\in A\}$.