# Few subsequence sums in Z_n x Z_n

 Importance: Medium ✭✭
\begin{conjecture} For every $0 \le t \le n-1$, the sequence in ${\mathbb Z}_n^2$ consisting of $n-1$ copes of $(1,0)$ and $t$ copies of $(0,1)$ has the fewest number of distinct subsequence sums over all zero-free sequences from ${\mathbb Z}_n^2$ of length $n-1+t$. \end{conjecture}
Definition: Given a sequence $\bf a$ of elements from an additive abelian group, we call a subsequence sum any group element expressable as a sum of some nontrivial subsequence of $\bf a$. We say that $\bf a$ is zero-free if $0$ is not a subsequence sum.
It is easy to see that every sequence $a_1,\ldots,a_n$ of elements from ${\mathbb Z}_n$ has a nontrivial subsequence which sums to zero (actually this holds for every group of order $n$). Just consider the elements $a_1$, $a_1 + a_2$, $\ldots$, $a_1 + \ldots, a_n$. If these elements are distinct, we have a zero sum. Otherwise, we have $a_1 + \ldots + a_j = a_1 + \ldots + a_k$ for some $1 \le j < k \le n$, but then $a_{j+1} + a_{j+2} + \ldots a_k = 0$. The same argument shows that whenever $0 \le t \le n-1$, every zero-free sequence of $t$ elements of ${\mathbb Z}_n$ must have at least $t$ distinct subsequence sums. In other words, the sequence consisting of $t$ copies of $1$ has the fewest number of distinct subsequence sums over all zero-free sequences in ${\mathbb Z}_n$ of length $t$.
In the group ${\mathbb Z}_n^2$, a theorem of Olsen shows that every sequence of length $\ge 2n-1$ has a nontrivial subsequence which sums to zero. However, we do not know what the minimum number of distinct subsequence sums is for a zero-free sequence of a given length. The above conjecture would appear to be the natural optimum.