Quotient and product sets of thin subsets of the positive integers

We study the cardinalities of $A/A$ and $AA$ for thin subsets $A$ of the set of the first $n$ positive integers. In particular, we consider the typical size of these quantities for random sets $A$ of zero density and compare them with the sizes of $A/A$ and $AA$ for subsets of the shifted primes and the set of sums of two integral squares.