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New Sum-Product Estimates for Real and Complex Numbers

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Abstract

A variation on the sum-product problem seeks to show that a set which is defined by additive and multiplicative operations will always be large. In this paper, we prove new results of this type. In particular, we show that for any finite set \(A\) of positive real numbers, it is true that

$$\begin{aligned} \Big |\Big \{\frac{a+b}{c+d}:a,b,c,d\in {A}\Big \}\Big |\ge {2|A|^2-1}. \end{aligned}$$

As a consequence of this result, it is also established that

$$\begin{aligned} |4^{k-1}A^{(k)}|:=|\underbrace{\underbrace{A\ldots {A}}_{k\,\,\text {times}}+\cdots {+A\ldots {A}}}_{4^{k-1} \mathrm{times}}|\ge {|A|^k}. \end{aligned}$$

Later on, it is shown that both of these bounds hold in the case when \(A\) is a finite set of complex numbers, although with smaller multiplicative constants.

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Notes

  1. The bound in (5) is the content of Proposition 4 in [8], and the bulk of that paper is devoted to proving this.

  2. We can avoid any potential issues arising from division by zero by simply assuming that \(A\) is a set of positive reals, and this will only change the implied constant in the eventual result.

  3. The number of lines \(P+P\) determines is just the number of lines through the origin which are needed in order to cover \(P+P\).

  4. The magnitude of \(q=(q_1,q_2)\) is the value \(|q|=\sqrt{q_1^2+q_2^2}\).

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Acknowledgments

The authors are very grateful to Misha Rudnev for his suggestion of extending these results to the complex setting. We are also grateful to the anonymous referees for their helpful suggestions. Antal Balog was partially supported by HNSF Grants K104183 and K109789. Oliver Roche-Newton was partially supported by Grant ERC-AdG. 321104 and EPSRC Doctoral Prize Scheme (Grant Ref: EP/K503125/1).

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Authors and Affiliations

  1. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, H-1053, Hungary

    Antal Balog

  2. Johann Radon Institute of Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, 4040, Linz, Austria

    Oliver Roche-Newton

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  1. Antal Balog
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  2. Oliver Roche-Newton
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Correspondence to Oliver Roche-Newton.

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Balog, A., Roche-Newton, O. New Sum-Product Estimates for Real and Complex Numbers. Discrete Comput Geom 53, 825–846 (2015). https://doi.org/10.1007/s00454-015-9686-x

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  • DOI: https://doi.org/10.1007/s00454-015-9686-x

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