# A problem in additive number theory.

Original Problem: Counterexample given below by user francis-jamet.

Let $A\subset \mathbb Z_n$ for some $n\in \mathbb{N}$.

If $A-A=\mathbb Z_n$, then $0\in A+A+A$

New Problem: Is the following statement true? If not, please give a counterexample.

If $A-A=\mathbb Z_n$ and $0\not\in A+A$, then $0\in A+A+A$.

• A small observation: if the statement is true, then $A+A+A$ also contains all multiples of three. – Colin McQuillan Jun 27 '12 at 16:26
• There are no counterexamples for $n \leq 16$. Link to a very naive Haskell program – sdcvvc Jun 27 '12 at 18:18
• All tests pass $A - A = {\mathbb Z}_n \implies A + A + A = {\mathbb Z}_n$. The stronger conclusion $A + A = \mathbb {\mathbb Z}_n$ is false: $n=6$ and $A=\{1,2,4\}$. – sdcvvc Jun 27 '12 at 19:10
• @sdcvvc Ha! No wonder I had a hard time proving it. (The false result I mean.) – user940 Jun 27 '12 at 19:27
• I've restored the original problem to go side by side with new problem. I think it is less confusing for readers this way. – user940 Jun 28 '12 at 14:08

For the original problem, there is a counterexample for $n=24$ and $A=\{3,9,11,15,20,21,23\}$.

There are no counterexamples for $n \leq 23$.

For the new problem, there is a counterexample:

$n=29$ and $A=\{4,5,6,9,13,22,28\}$.

There are no counterexamples for $n \leq 28$.

• A test in Python (it gives "True"): a=[3,9,11,15,20,21,23]; set([(x-y+24) % 24 for x in a for y in a])==set(range(24)) – Colin McQuillan Jun 28 '12 at 9:08
• Oh, and the following expression evaluates to False: 0 in set([(x+y+z) % 24 for x in a for y in a for z in a]) – Colin McQuillan Jun 28 '12 at 9:09
• Thanks very much for the counterexample!! – Terry Zhou Jun 28 '12 at 13:34
• @TerryZhou: So as not to undermine francis-jamet's efforts in solving your original problem, I think it would make more sense to start a new question and include a link to this one. – Cam McLeman Jun 28 '12 at 14:29
• I just finished writing a program to search for counterexamples to both problems as francis updated this. So I'll just leave my results as a comment. I found another counterexample to the original problem [1,3,4,9,13,15,21]. And to the second problem I found [1,7,16,20,23,24,25]. – JSchlather Jun 28 '12 at 18:15