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The Erdős-Ginzberg-Ziv theorem with units. (English) Zbl 1206.11032

Author’s summary: “Let \(x_1,\dots ,x_r\) be a sequence of elements of \(\mathbb Z_n\), the integers modulo \(n\). How large must \(r\) be to guarantee the existence of a subsequence \(x_{i_1},\dots ,x_{i_n}\) and units \(\alpha_1,\dots ,\alpha_n\) with \(\alpha_1x_{i_1}+\dots +\alpha_nx_{i_n}=0\)? The author’s main aim in this paper is to show that \(r=n+a\) is large enough, where \(a\) is the sum of the exponents of primes in the prime factorization of \(n\). This result, which is best possible, could be viewed as a unit version of the Erdős-Ginzberg-Ziv theorem. This proves a conjecture of S. D. Adhikari, Y. G. Chen, J. B. Friedlander, S. V. Konyagin, and F. Pappalardi [Discrete Math. 306, No. 1, 1–10 (2006; Zbl 1161.11311)].
He also discusses a number of related questions, and make conjectures which would greatly extend a theorem of Gao.”
Reviewer’s remark: The main result of the paper was also proved independently by F. Luca [Discrete Math. 307, No. 13, 1672–1678 (2007; Zbl 1123.11012)] using a somewhat different method.

MSC:

11B50 Sequences (mod \(m\))
11B75 Other combinatorial number theory
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References:

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