If $S,T$ are two sets of integers, define $S*T$ to be the set $S*T = \{st \mid s \in S, t \in T\}$. Let $[1,n]$ denote the set of integers in the range from $1$ to $n$, i.e., $[1,n] = \{1,2,3,\dots,n\}$.
Define $f(n)$ to be the cardinality of $U_n = [1,n] * [1,n]$. Is there a reasonable asymptotic estimate for $f(n)$? Is it $f(n) = c \cdot n^2 + o(n^2)$ for some constant $c$, and if so, is there any rough estimate for the constant $c$? In other words: how many numbers less than or equal to $n^2$ can be written as the product of two numbers less than or equal to $n$?
Define $g(n)$ to be the cardinality of $V_n = [1,n] * [1,n] * [1,n]$. Is there a reasonable asymptotic estimate for $g(n)$? Is it $c' \cdot n^3 + o(n^3)$, and if so, what's $c'$ (roughly)?
I can see that $f(n) \le n^2/2 + o(n^2)$ and $g(n^3) \le n^3/6 + o(n^3)$, by symmetry, but I don't have any better bound. I don't have a lower bound, either.
What I tried: I can see that any integer that contains a prime factor larger than $n$ cannot be in $U_n$. Defining
$$W_n= \{pk \mid n<p<n^2, p \text{ is prime}, 1\le k \le n^2/p\},$$
it follows that $W_n \subseteq \overline{U_n}$. However, this didn't seem to help. I tried crudely estimating the density of $\overline{U_n} \cap [jn,jn+n-1]$ for some integer $j\ge 1$ and then summing $j=1,2,\dots,n-1$, but my estimate was too crude: using the density of primes, I got the estimate $|\overline{U_n} \cap [jn,jn+n-1]| \approx n/{j \log(jn)}$, but summing this looks ugly. I can upper-bound $n/{j \log(jn)}$ by $n/{j \log n}$, and then the sum is $H_{n-1} n^2/\log n$, but this is roughly $n^2$, which isn't helpful. Dead end.
Let $A = \{1,2,\ldots,N\}$. It was proved in 2008 by Kevin Ford that $$ |A*A| \asymp \frac{N^2}{(\log N)^c (\log\log N)^{3/2}} , $$ where $c = 1 - \frac{1+\log\log 2}{\log 2}$. Here is a reference:
What I stated above is only a special case of the results proved in his paper.