TITLE: Trisecting $3$-fold sumsets: is the middle part always thick?
QUESTION [6 upvotes]: 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.

REPLY [6 votes]: 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 $101, 53, 13$.