Here is a truly minimalistic and seemingly basic question which should have a simple solution (I hope it does).


Let $A$ be a finite set of integers with the smallest element $0$ and the largest element $l$. The sumset $C:=3A$ resides in the interval $[0,3l]$, and we let $C\_1:=C\cap[0,l]$, $C\_2:=C\cap[l,2l]$, and $C\_3:=C\cap[2l,3l]$.



> 
> Is it true that $|C\_2|\ge\frac12\,(|C\_1|+|C\_3|)$, for any choice of the set $A$?
> 
> 
> 


Computations seem to suggest that the answer is in the affirmative.



Trisecting $3$-fold sumsets: is the middle part always thick?
No. Take $A = \{0,1,\ldots,9,10,20,30,\ldots,90,100,200,300,\ldots,900,1000\}$. Then $|C\_1|=1001$, $|C\_2|=272$ and $|C\_3|=29$.


A smaller counterexample in the same spirit is $\{0,1,2,3,4,5,10,15,20,25,50,75,100\}$, with sizes $101, 53, 13$.


