On the sum of dilations of a set

Antal Balog, George Shakan

Type: Article

Publication Date: 2014-01-01

Citations: 12

DOI: https://doi.org/10.4064/aa164-2-5

Abstract

We show that for any relatively prime integers $1\leq p< q$ and for any finite $A \subset \mathbb {Z}$ one has $$|p \cdot A + q \cdot A | \geq (p + q) |A| - (pq)^{(p+q-3)(p+q) + 1}.$$

Locations

Acta Arithmetica - View - PDF

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Action	Title	Year	Authors
+ PDF Chat	On direct and inverse problems related to some dilated sumsets	2024	Ramandeep Kaur Sandeep Singh
+	Sumset problem on dilated sets of integers	2024	Sandeep Singh Ramandeep Kaur Mahendra Kumar Verma
+	On a sumset problem for dilated integer sets	2024	Ramandeep Kaur Jagannath Bhanja Sandeep Singh
+	On the size of $A+\lambda A$ for algebraic $\lambda$	2020	Dmitry Krachun Fedor Petrov
+ PDF Chat	Sums of transcendental dilates	2023	David Conlon Jeck Lim
+ PDF Chat	On the size of A+λA for algebraic λ	2023	Dmitry Krachun Fedor Petrov
+	Sums of linear transformations in higher dimensions	2019	Akshat Mudgal
+ PDF Chat	Sums of Linear Transformations in Higher Dimensions	2019	Akshat Mudgal
+ PDF Chat	Sum of Many Dilates	2015	George Shakan
+	On Sumset Problems and Their Various Types	2022	Ramandeep Kaur Sandeep Singh

Works Cited by This (7)

Action	Title	Year	Authors
+ PDF Chat	On sums and products of integers	1983	Péter L. Erdős Endre Szemerédi
+ PDF Chat	On the number of sums and products	1997	György Elekes
+	A note on sum-product estimates	2011	Antal Balog
+	Bounding multiplicative energy by the sumset	2009	József Solymosi
+ PDF Chat	A Lower Bound for the Size of a Minkowski Sum of Dilates	2010	Yahya Ould Hamidoune Juanjo Rué
+ PDF Chat	On a Sumset Problem for Integers	2014	Shanshan Du Hui-Qin Cao Zhi‐Wei Sun
+	A lower bound for the size of the sum of dilates	2014	George Shakan