Is there a finite set $S$ of real numbers bigger than $1$,
and a positive real number $\alpha$ such that if
$P$ = set of finite products, (allow duplication) of elements in $S$
$Q$ = set of differences of two elements in $P$
then, there are infinitely many elements in Q, less than $\alpha$.
I guess this problem must have been studied.
I don't have skill's to attack this problem, but I am just so curious about the result.
For example, if $S$ is $\{2,3\}$, then $P$ becomes set of integers of a form $ 2^m 3^n$
where $m,n$ are non-negative integers.