On the discretized sum-product problem

Larry Guth; Nets Hawk Katz; Joshua Zahl

doi:10.1093/imrn/rnz360

On the discretized sum-product problem

Classical Analysis and ODEs 2023-08-24 v3 Combinatorics Metric Geometry

Authors: Larry Guth , Nets Hawk Katz , Joshua Zahl

View on arXiv ↗ PDF ↗ DOI ↗

Abstract

We give a new proof of the discretized ring theorem for sets of real numbers. As a special case, we show that if $A\subset\mathbb{R}$ is a $(\delta,1/2)_1$ -set in the sense of Katz and Tao, then either $A+A$ or $A.A$ must have measure at least $|A|^{1-\frac{1}{68}}$

Keywords

arithmetic progressions additive number theory divisor function

Cite

@article{arxiv.1804.02475,
  title  = {On the discretized sum-product problem},
  author = {Larry Guth and Nets Hawk Katz and Joshua Zahl},
  journal= {arXiv preprint arXiv:1804.02475},
  year   = {2023}
}

Comments

13 pages, 0 figures. v3: final version, to appear in IMRN

On the discretized sum-product problem

Abstract

Keywords

Cite

Comments

Related papers