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arXiv · 1703.09549

Variations on the sum-product problem II

Brendan Murphy·Oliver Roche-Newton·Ilya Shkredov
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Abstract

This is a sequel to the paper arXiv:1312.6438 by the same authors. In this sequel, we quantitatively improve several of the main results of arXiv:1312.6438, and build on the methods therein. The main new results is that, for any finite set $A \subset \mathbb R$, there exists $a \in A$ such that $|A(A+a)| \gtrsim |A|^{\frac{3}{2}+\frac{1}{186}}$. We give improved bounds for the cardinalities of $A(A+A)$ and $A(A-A)$. Also, we prove that $|\{(a_1+a_2+a_3+a_4)^2+\log a_5 : a_i \in A \}| \gg \frac{|A|^2}{\log |A|}$. The latter result is optimal up to the logarithmic factor.

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Brendan Murphy, Oliver Roche-Newton, Ilya Shkredov. 2017-04-04. Variations on the sum-product problem II. https://arxiv.org/abs/1703.09549

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