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Statement

Notation (pp. 2--3): a sequence over $G$ is an element of the free abelian monoid $\mathcal F(G)$, that is, a multiset of elements of $G$; $|S|$ is its length, $\mathrm{supp}(S)$ the set of distinct terms, $g^k$ the sequence of $k$ copies of $g$, and $\Sigma_r(S)$ the set of sums of the subsequences of $S$ of length $r$. The paper writes $[a,b]$ for the set of integers from $a$ to $b$ without defining it.

Theorem 3.4 (p. 5, first part quoted). "Let $G$ be a finite abelian group of order $n$ and let $S\in\mathcal F(G)$ with $|S|=n$. Suppose there is a unique $r\in[1,n]$ such that $0\in\Sigma_r(S)$. Then $|\mathrm{supp}(S)|\le2$."

The theorem goes on (p. 5) to say which forms $S$ can then take.

• $G$ non-cyclic. Then $G=\langle h\rangle\oplus\langle g\rangle\cong C_2\oplus C_{2m}$, $r=\frac n2=2m$, and $S$ is $g^{n-1}g'$, or $g^{n/2+x}(h+g)^{n/2-x}$, or $g^{n/2+x}(h+\frac{n+4}4g)^{n/2-x}$, where $g\in G$, $h,g'\in G\setminus\langle g\rangle$, $\mathrm{ord}(g)=\frac n2$, $\mathrm{ord}(h)=2$ and $x\in[1,\frac n2-1]$ is odd.
• $G$ cyclic. Then there is a generator $g$ of $G\cong C_n$ such that one of the following holds: $S=g^{n-1}g'$ for some $g'\in G$; or $S=(2g)^{n-1}g''$ for some $g''\in G\setminus\langle2g\rangle$; or $n$ is odd, $r=\frac{n+1}2$ and $S=g^{n-2}(\frac{n+1}2g)^2$; or $n\equiv2\pmod 4$, $r=\frac n2$ and $S=(2g)^{n/2+x}(\frac{n+4}2g)^{n/2-x}$ with $x\in[0,\frac n2-1]$ even; or $n$ is even, $r=\frac n2$ and $S=g^{n/2+x}(\frac{n+2}2g)^{n/2-x}$ with $x\in[0,\frac n2-1]$ and $\frac n2-x$ odd.

The theorem states these forms as necessary; it does not state that every sequence of these forms has a unique zero-sum length. The paragraph before the theorem (p. 5) adds that most non-cyclic groups admit no sequence meeting the hypotheses, since $2\mathsf D(G)\le|G|$ holds for most of them, $\mathsf D(G)$ being the Davenport constant (p. 3).

The hypothesis read as Graham's. Since $\mathsf D(G)\le|G|$ (p. 3), a sequence of $n$ terms always has a nonempty zero-sum subsequence, so the hypothesis says exactly that all nonempty zero-sum subsequences of $S$ have the same length (an observation of this page). With $G=C_p$ for a prime $p$ the theorem is Graham's conjecture as the paper states it (Conjecture 1.1, p. 1), the term $0$ not excluded.

Source. D. J. Grynkiewicz, Note on a conjecture of Graham, European J. Combin. 32 (2011), no. 8, 1336--1344, doi:10.1016/j.ejc.2011.06.004, read in the arXiv preprint arXiv:0903.3200v1 (18 March 2009) identified on the source card, whose pagination and labels are used here; the journal text was not compared. Theorem 3.4 on p. 5, its proof on pp. 6--10, the remark on the prime case on p. 10.

Read depth. Claims checked: the statement and the list of forms were read clause by clause on the page images. The proof was read through for its structure, not checked step by step. Nothing here is independently reviewed.

Proof pointer

Pp. 6--10, in four steps, with $g$ a term of maximal multiplicity $l=\mathsf h(S)$. The cases $r=1$ and $r=n$ are immediate, so $1<r<n$ and $0\notin\mathrm{supp}(S)$. Step 1 (p. 6) shows that $l\ge\max\{r,n-r+1\}$, or that $n$ is even and $S$ consists of two terms of order $n$ each taken $\frac n2$ times; it applies Theorem 2.2 (a case of the DeVos--Goddyn--Mohar theorem) to the sums of $n-r-1$ or $r-1$ terms of $S$ padded with zeros. Step 2 (p. 7) treats $\mathrm{ord}(g)<n$ through the Davenport constant of $G/\langle g\rangle$ and Lemma 3.3, which leaves $G$ cyclic with generator $g$. Step 3 (pp. 7--8) uses Lemma 3.1 to produce a sum of more than $r$ terms outside the multiples of $g$ already covered. Step 4 (pp. 9--10) applies Proposition 2.1(ii) to a sumset $A+B=G$ built from multiples of $g$ and the partial sums of the remaining terms, and finishes with Lemma 3.2. The remark after the proof (p. 10) shortens it for $G=C_p$ with $p$ prime: Step 2 is not needed, $n$ odd removes the extra case of Step 3, and Step 1 follows from the Cauchy--Davenport theorem in place of DeVos--Goddyn--Mohar.

Dependencies

Proposition 2.1 (representation counts in sumsets, cited from Geroldinger--Halter-Koch and Nathanson), Theorem 2.2 (a special case of the DeVos--Goddyn--Mohar theorem), the Cauchy--Davenport theorem and a set-partition result of Bialostocki, Dierker, Grynkiewicz and Lotspeich (their Proposition 2.1) for the prime case, the bound $\mathsf D(G)\le|G|$, and Lemmas 3.1--3.3 of the paper (pp. 4--5), all taken at statement level here.

Bears on

• Problem 541: with $G=C_p$ and $S$ the sequence $a_1,\ldots,a_p$, the problem's hypothesis that every nonempty zero-sum set of indices has one size $r$ is the theorem's hypothesis, and the conclusion $|\mathrm{supp}(S)|\le2$ is the problem's answer, for every prime $p$ and with the residue $0$ admitted; with $G=C_n$ it gives the same statement for every modulus $n$.