On the length of arithmetic progressions in linear combinations of \(S\)-units
Let \(K\) be an algebraically closed field of characteristic \(0\), \(t\) a positive integer, \({\mathcal A}\) a finite subset of \(K^t\), and \(\Gamma\) a subgroup of the multiplicative group \(K^*\), of finite rank. For a positive integer \(t\), denote by \({\mathcal H}_t\) the set of sums \(\sum_{i=1}^t a_ix_i\), with \((a_1,\dots , a_t)\in{\mathcal A}\), and \(x_1,\dots , x_t\in\Gamma\). Denote by \(L\) the maximum of the lengths of the arithmetic progressions contained in \({\mathcal H}_t\). Recently, Hajdu and independently in a special case Jarden and Narkiewicz proved that \(L\) is bounded above by a quantity depending only on the cardinality \(n\) of \({\mathcal A}\), the rank \(r\) of \(\Gamma\), and \(t\). In their proofs they used a result of \textit{H. P. Schlickewei}, \textit{W. M. Schmidt} and the reviewer [Ann. Math. (2) 155, No. 3, 807--836 (2002; Zbl 1026.11038)] on the number of solutions of linear equations with unknowns from \(\Gamma\), as well as \textit{B. L. van der Waerden}'s theorem on monochromatic arithmetic progressions [Nieuw Arch. 15, 212--216 (1927; JFM 53.0073.12)]. In the present paper, the authors give a new proof of this result, with an explicit bound \(L<\exp\big\{ \big( 8(n+t+r)\big)^{8(n+t+r)^4}\big\}\). In their new proof, the authors remove the application of van der Waerden's theorem which led to a dramatic improvement of the upper bound. Further, they use the recent sharpening by \textit{F. Amoroso} and \textit{E. Viada} [Duke Math. J. 150, No. 3, 407--442 (2009; Zbl 1234.11081)] of the result of Schlickewei, Schmidt and the reviewer. Editorial note: An erratum to this paper correcting Lemma 2.1 and Theorem 1.1 has been published in ibid. 103, No. 4, 399--400 (2014; Zbl 1304.11017).
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