An update on the sum-product problem in $\mathbb{R}$

Appears in collection : Additive Combinatorics / Combinatoire additive

Discussing recent work joint with M. Rudnev [2], I will discuss the modern approach to the sum-product problem in the reals. Our approach builds upon and simplifies the arguments of Shkredov and Konyagin [1], and in doing so yields a new best result towards the problem. We prove that $max(\left | A+A \right |,\left | A+A \right |)\geq \left | A \right |^{\frac{4}{3}+\frac{2}{1167}-o^{(1)}}$ , for a finite $A\subset \mathbb{R}$. At the heart of our argument are quantitative forms of the two slogans ‘multiplicative structure of a set gives additive information’, and ‘every set has a multiplicatively structured subset’.

Information about the video

Date of recording 07/09/2020
Date of publication 29/09/2020
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.19654203
Cite this video Stevens, Sophie (07/09/2020). An update on the sum-product problem in $\mathbb{R}$. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19654203
URL https://dx.doi.org/10.24350/CIRM.V.19654203

Domain(s)

Bibliography

[1] Konyagin, S.V., and I.D. Shkredov, On sum sets of sets having small product set, Proceedings of the Steklov Institute of Mathematics 290.1 (2015): 288-299. - https://doi.org/10.1134/S0081543815060255
[2] Rudnev, M, and S. Stevens, An update on the sum-product problem, arXiv preprint arXiv:2005.11145 (2020). - https://arxiv.org/abs/2005.11145