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**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

It is established that for any finite set of positive real numbers A, we have |A/A+A| >> |A|^{3/2+1/26} / log^{5/6}|A|.

Oliver Roche-Newton. An Improved Bound for the Size of the Set A/A+A. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 69:1-69:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{rochenewton:LIPIcs.SoCG.2018.69, author = {Roche-Newton, Oliver}, title = {{An Improved Bound for the Size of the Set A/A+A}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {69:1--69:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.69}, URN = {urn:nbn:de:0030-drops-87820}, doi = {10.4230/LIPIcs.SoCG.2018.69}, annote = {Keywords: sum-product estimates, expanders, incidence theorems, discrete geometry} }

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**Published in:** LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

In this note it is established that, for any finite set A of real numbers, there exist two elements a, b from A such that |(a + A)(b + A)| > c|A|^2 / log |A|, where c is some positive constant. In particular, it follows that |(A + A)(A + A)| > c|A|^2 / log |A|. The latter inequality had in fact already been established in an earlier work of the author and Rudnev, which built upon the recent developments of Guth and Katz in their work on the Erdös distinct distance problem. Here, we do not use those relatively deep methods, and instead we need just a single application of the Szemerédi-Trotter Theorem. The result is also qualitatively stronger than the corresponding sum-product estimate from the paper of the author and Rudnev, since the set (a + A)(b + A) is defined by only two variables, rather than four. One can view this as a solution for the pinned distance problem, under an alternative notion of distance, in the special case when the point set is a direct product A x A. Another advantage of this more elementary approach is that these results can now be extended for the first time to the case when A is a set of complex numbers.

Oliver Roche-Newton. A Short Proof of a Near-Optimal Cardinality Estimate for the Product of a Sum Set. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 74-80, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{rochenewton:LIPIcs.SOCG.2015.74, author = {Roche-Newton, Oliver}, title = {{A Short Proof of a Near-Optimal Cardinality Estimate for the Product of a Sum Set}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {74--80}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.74}, URN = {urn:nbn:de:0030-drops-51200}, doi = {10.4230/LIPIcs.SOCG.2015.74}, annote = {Keywords: Szemer\'{e}di-Trotter Theorem, pinned distances, sum-product estimates} }

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