This is a refined version of the [question I asked yesterday](https://mathoverflow.net/q/394059/9924).


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 I write $C\_1:=C\cap[0,l)$, $C\_2:=C\cap[l,2l]$, and $C\_3:=C\cap(2l,3l]$.



> 
> Is it true that $|C\_2|\ge\min\{|C\_1|,|C\_3|\}$, for any choice of the set $A$?
> 
> 
> 


A brute force computation shows that there are no counterexamples with $l\le 22$.



Trisecting $3$-fold sumsets, II: is the middle part ever thin?
The title and body are asking the question in opposite senses. For the title, the answer is "yes" (it can be thin), and for the body, the answer if "no" (it is not true that it is never thin).


An example is
$$
A = \{0,1,2,3,8,11,26,38,56,69,85,89,179,189,197,221,226,243,254,257,264,266,269,270\}
$$
where the sumset parts $C\_1,C\_2,C\_3$ have sizes $270, 268, 269$.


Perhaps a word about the construction. The underlying idea was to take two shorter sets $P,Q$ that are as thin (small) as possible, while their sumsets $3P, 3Q$ would cover some interval $[0,b]$ fully. Then make $A = P \cup (l-Q)$ with some conveniently chosen $l$. Hopefully then $3A$ covers the left and right thirds almost fully, while the middle third *might* have enough holes to make a counterexample. Note the "might" – this is all heuristic, and it is not done until you just try it out.


I took $$P=\{0,1,2,3,8,11,26,38,56,69,85,89\}$$ $$Q=\{0,1,4,6,13,16,27,44,49,73,81,91\}$$ from Challis and Robinson (2010), [Some Extremal Postage Stamp Bases](https://cs.uwaterloo.ca/journals/JIS/VOL13/Challis/challis6.html), *Journal of Integer Sequences* 13, Article 10.2.3; Table "h=3" on page 9. Here $P$ and $Q$ are minimal-cardinality additive bases (of order 3) for the interval $[0,186]$, so that part of the construction is done.


Then a brute force search over some possible values of $l$ gave a hit at $l=270$. Indeed, $C\_1$ has no "holes" (it covers the integer interval $[0,269]$ fully). $C\_2$ has three holes in interval $[270,540]$, and $C\_3$ has one hole in interval $[541,810]$.


