$4$ vectors in $\mathbb{Z_4}^9$that sum to $0$.

Question

Problem: Given $2018$ elements not necessarily distinct from $\mathbb{Z_4}^9=\mathbb{Z_4} \times \mathbb{Z_4}\times \mathbb{Z_4}\times \mathbb{Z_4}\times \mathbb{Z_4}\times \mathbb{Z_4}\times \mathbb{Z_4}\times \mathbb{Z_4}\times \mathbb{Z_4}$, Prove that there exist 4 elements from these $2018$ which sum to $\text{0}= (0,0,0,0,0,0,0,0,0)$. Obviously you cant use an element more than the number of times it is available.

Origin: This problem originates from a 'number theory' question: Given $2018$ integers all of whose prime factors are $\leq 23$ show that there are $4$ whose product is a fourth power.

My attempt: The only slightly fruitful approach I had was to attach this problem coordinate-wise.. Since, there are 2018. There will be an integer in $\mathbb{Z_4}$ that occurs at least $2018/4$ that is $505$ times. Now starting with these $505$ we go to the next coordinate and so on. But in this way I will need about $4^9 > 2018$ integers to do my job.

I feel a more global approach is needed. I tried some contradiction but that also didnt bear any fruit.

Please help. Thanks.

Answer 1

In my opinion this problem is more about pigeonhole-principle rather than number-theory.

Anyway, a pigeonhole exercise of this type can often be done in two stages. In the first we produce a lot of squares. Here we would first produce a lot of pairs of vectors belonging to $\{0,2\}^9$.

Observation. Given $513$ vectors from $\Bbb{Z}_4^9$ we can find among them two vectors $u$ and $v$ such that their sum has only even components. In other words $u+v\in\{0,2\}^9$.

Proof. This is basic pigeon-holing. There are $2^9=512$ parity combinations, so we can find $u,v$ such that $u_i\equiv v_i\pmod2$ for all $i$. This pair works.

Then we can proceed with the main task. First run the following algorithm:

1. Initialize $S$ to be the list of the given $2018$ vectors. Initialize a counter $n=0$, and initialize another list $L$ to be empty.
2. If more than $512$ elements remain in the list, the observation tells us that we can find a pair of entries $u,v$ from the list $S$ such that $u+v\in\{0,2\}^9$. Otherwise, the algorithm stops here.
3. Increment $n\leftarrow n+1$, set $u_n=u,v_n=v$. Remove $u$ and $v$ from the list $S$ and add the pair $(u_n,v_n)$ to the list L.
4. Go to step 2.

After this algorithm has run its course, there are at most $512$ vectors remaining in $S$. Therefore we have removed at least $2018-512=1506$ elements from the list $S$. Thus we can conclude that the list $L$ now has $n=753>512$ pairs $(u_i,v_i),i=1,2,\ldots,n$ such that for all $i$ we have $u_i+v_i\in\{0,2\}^9$.

We are now well places to reapply the observation to the collection of $753$ vectors $(u_i+v_i)/2$. The observation gives us pairs $(u_i,v_i)$, $(u_j,v_j)$, $i\neq j$, such that $(u_i+v_i+u_j+v_j)/2\in\{0,2\}^9$.

Therefore all the components of $u_i+v_i+u_j+v_j$ vanish modulo four.

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Strange. Looks like I didn't need nearly all of the $2018$ vectors, $1537$ would seem to suffice to produce $513$ pairs?