On the size of the set AA+A

Oliver Roche‐Newton, Imre Z. Ruzsa, Chun‐Yen Shen, Ilya D. Shkredov

Type: Article

Publication Date: 2018-09-21

Citations: 7

DOI: https://doi.org/10.1112/jlms.12177

Abstract

It is established that there exists an absolute constant $c>0$ such that for any finite set $A$ of positive real numbers $$|AA+A| \gg |A|^{\frac{3}{2}+c}.$$ On the other hand, we give an explicit construction of a finite set $A \subset \mathbb R$ such that $|AA+A|=o(|A|^2)$, disproving a conjecture of Balog.

Locations

Journal of the London Mathematical Society - View
Repository of the Academy's Library (Library of the Hungarian Academy of Sciences) - View - PDF
arXiv (Cornell University) - View - PDF
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+	Non-commutative methods in additive combinatorics and number theory	2021	Ilya D. Shkredov
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Works Cited by This (13)

Action	Title	Year	Authors
+	Convexity and Sumsets	2000	György Elekes Melvyn B. Nathanson Imre Z. Ruzsa
+	A note on sum-product estimates	2011	Antal Balog
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+ PDF Chat	Variations on the Sum-Product Problem II	2017	Thomas Brendan Murphy Oliver Roche‐Newton Ilya D. Shkredov
+	On discrete values of bilinear forms	2015	Alex Iosevich Oliver Roche‐Newton Misha Rudnev
+	New results on sums and products in ℝ	2016	Sergeĭ Konyagin Ilya D. Shkredov
+ PDF Chat	Some remarks on the asymmetric sum-product phenomenon	2018	Ilya D. Shkredov
+	Additive Combinatorics	2006	Terence Tao Van H. Vu
+ PDF Chat	Variations on the Sum-Product Problem	2015	Thomas Brendan Murphy Oliver Roche‐Newton Ilya D. Shkredov
+ PDF Chat	On a question of A. Balog	2016	Ilya D. Shkredov