Vol. 9, No. 1, 2020
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Discretized sum-product for large sets

Changhao Chen
Vol. 9 (2020), No. 1, 17–27
DOI: 10.2140/moscow.2020.9.17
Abstract

Let A ⊂ [1,2] be a (δ,σ)-set with measure |A| = δ1−σ in the sense of Katz and Tao. For σ ∈(1 2,1) we show that

|A + A| + |AA|≿ δ−c|A|

for c = (1 − σ)(2σ − 1)∕(6σ + 4). This improves the bound of Guth, Katz, and Zahl for large σ.

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Keywords
$(\delta,\sigma)$-sets, sum set estimates, Fourier transform
Mathematical Subject Classification 2010
Primary: 05B99
Milestones
Received: 17 June 2019
Revised: 3 October 2019
Accepted: 28 October 2019
Published: 8 January 2020
Authors
Changhao Chen
School of Mathematics and Statistics
University of New South Wales
Sydney, NSW
Australia