On a conjecture of Kemnitz (Q2746250)

Let \(f(n,d)\) denote the least number such that any sequence of \(f(n,d)\) lattice points in \(\mathbb{Z}^d\) contains a subsequence of size \(n\) whose sum is \(0\). By the pigeonhole principle it follows that NEWLINE\[NEWLINE2^d(n-1)+1 \leq f(n,d) \leq n^d(n-1)+1.NEWLINE\]NEWLINE \textit{A. Kemnitz} [Ars. Comb. 16B, 151-160 (1983; Zbl 0539.05008)] conjectured that \(f(n,2)=4n-3\). It is sufficient to prove this for prime \(n\) since the full conjecture then follows by multiplicativity. \textit{L. Rónyai} [Combinatorica 20, 569-573 (2000; Zbl 0963.11013)] proved that \(f(p,2) \leq 4p-2\) which implies that \(f(n,2) \leq 4.1 n\). \textit{W. D. Gao} [J. Comb. Theory, Ser. A 95, 387-389 (2001; Zbl 0992.11027)] extended this to powers of primes.NEWLINENEWLINENEWLINEThe paper under review proves: a sequence of \(4n-3\) elements in \(\mathbb{Z}^2\), where one element occurs at least \(\lfloor \frac{n}{2}\rfloor\) times, has a zero sum of length \(n\). Moreover, \(f(nm,2)=4nm-3\) holds when \(f(n,2)=4n-3\) and \(n \geq (2m^3-3m^2+3)/(4m)\) for \(m \geq 2\). This improves on the work of \textit{W. D. Gao} [J. Number Theory 61, 97-102 (1996; Zbl 0870.11016)].