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A new expander and improved bounds for \(A(A+A)\)
The main result in this paper concerns a new five-variable expander. It is proven that for any finite set of real numbers \(A\), $$|\{(a_1+a_2+a_3+a_4)^2+\log a_5 :a_1,a_2,a_3,a_4,a_5 \in A \}| \gg \frac{|A|^2}{\log |A|}.$$ This bound is optimal, up to logarithmic factors. The paper also gives new l...
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Published in: | arXiv.org 2017-04 |
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Main Author: | |
Format: | Article |
Language: | English |
Subjects: | |
Online Access: | Get full text |
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Summary: | The main result in this paper concerns a new five-variable expander. It is proven that for any finite set of real numbers \(A\), $$|\{(a_1+a_2+a_3+a_4)^2+\log a_5 :a_1,a_2,a_3,a_4,a_5 \in A \}| \gg \frac{|A|^2}{\log |A|}.$$ This bound is optimal, up to logarithmic factors. The paper also gives new lower bounds for \(|A(A-A)|\) and \(|A(A+A)|\), improving on results from arXiv:1312.6438. The new bounds are $$|A(A-A)| \gtrapprox |A|^{3/2+\frac{1}{34}}$$ and $$|A(A+A)| \gtrapprox |A|^{3/2+\frac{5}{242}}.$$ |
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ISSN: | 2331-8422 |