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Sums and products of distinct sets and distinct elements in \(\mathbb C\). (English) Zbl 1227.11036

The sum-product phenomenon predicts that a finite set \(A\) in a ring \(R\) should have either a large sumset \(A+A\) or large product set \(A \cdot A\) unless it is in some sense “close” to a finite subring of \(R\). This phenomenon has been analysed intensively in recent years. In this paper, let \(A\) and \(B\) be two finite subsets of \(\mathbb{C}\) such that \(|B| = C|A|\), the author states the following variant of the sum product phenomenon: If \(|AB|<\alpha |A|\) and \(\alpha \ll \log |A|\), then \(|kA+lB|\gg|A|^k|B|^l\), by applying a result of Evertse, Schlickewei and Schmidt on linear equations with variables taking values in multiplicative groups of finite rank and an earlier theorem of Ruzsa about sumsets in \(\mathbb{R}^d\). Except for this, the author derives a lower bound on \(g_\mathbb{C}(n)\) , where \(g_\mathbb{C}(n)= \min_{A\subset \mathbb{C},\;|A|=n}\{|A^+|+|A^{\times}|\}\) with \(A^+\) and \(A^{\times}\) were the sums and products of distinct elements from a finite set \( A\subset \mathbb{C}\). This result generates Chang’s proof of the lower bound on \(g_\mathbb{Z}(n)\).

MSC:

11B13 Additive bases, including sumsets
11B30 Arithmetic combinatorics; higher degree uniformity
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