REAL

New Sum-Product Estimates for Real and Complex Numbers

Balog, Antal and Roche-Newton, O. (2015) New Sum-Product Estimates for Real and Complex Numbers. DISCRETE AND COMPUTATIONAL GEOMETRY, 53 (4). pp. 825-846. ISSN 0179-5376

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Abstract

A variation on the sum-product problem seeks to show that a set which is defined by additive and multiplicative operations will always be large. In this paper, we prove new results of this type. In particular, we show that for any finite set A of positive real numbers, it is true that |{a+bc+d:a,b,c,d∈A}|≥2|A|2-1.As a consequence of this result, it is also established that |4k-1A(k)|:=|A…A⏟ktimes+⋯+A…A⏟4k-1times|≥|A|k.Later on, it is shown that both of these bounds hold in the case when A is a finite set of complex numbers, although with smaller multiplicative constants. © 2015, Springer Science+Business Media New York.

Item Type: Article
Uncontrolled Keywords: Sum-product estimates; Elementary geometry; Complex numbers
Subjects: Q Science / természettudomány > QA Mathematics / matematika
SWORD Depositor: MTMT SWORD
Depositing User: MTMT SWORD
Date Deposited: 17 Feb 2016 05:35
Last Modified: 17 Feb 2016 05:35
URI: http://real.mtak.hu/id/eprint/33579

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