Variations on the Sum-Product Problem II

Brendan Murphy; Oliver Roche-Newton; Ilya Shkredov

doi:10.1137/17M112316X

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.

Original language	English
Pages (from-to)	1878-1894
Number of pages	17
Journal	SIAM Journal on Discrete Mathematics
Volume	31
Issue number	3
Early online date	20 Aug 2017
DOIs	https://doi.org/10.1137/17M112316X
Publication status	Published - 2017

Bibliographical note

This paper supersedes arXiv:1603.06827

Keywords

math.CO
math.NT

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10.1137/17M112316X

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https://arxiv.org/abs/1703.09549

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title = "Variations on the Sum-Product Problem II",

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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author = "Brendan Murphy and Oliver Roche-Newton and Ilya Shkredov",

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year = "2017",

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language = "English",

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T1 - Variations on the Sum-Product Problem II

AU - Murphy, Brendan

AU - Roche-Newton, Oliver

AU - Shkredov, Ilya

N1 - This paper supersedes arXiv:1603.06827

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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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KW - math.NT

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U2 - 10.1137/17M112316X

DO - 10.1137/17M112316X

M3 - Article (Academic Journal)

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EP - 1894

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 3

ER -

Variations on the Sum-Product Problem II

Abstract

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Keywords

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