I am interested in whether there are any conclusion/conterexample that the sum&difference of finite integer sets by more times is smaller than the sum&difference by less times.
The question I am actually thinking about is the following: Is there any example of finite integer set $U,V$, s.t. $U-V=\{1,2,...,m\}$, $U+U+V+V\neq\{1,2,...,m\}\pmod m$ for some m.
I think I knew nothing when comparing 4 sums and 2 difference, even the $U=V$ case or without modulo $m$ case.
when $U=\{x^2,x^2+1:x\in Z\}$, $V=\{x^2:x\in Z\}$, then $U-V=Z$, and $7*4^n \notin U+V+V$, this may show that 3 sums may be smaller than 2 difference? (though these are two infinite sets)
