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An update on the sum-product problem in $\mathbb{R}$
Virtualconference
Authors : Stevens, Sophie (Author of the conference)
CIRM (Publisher )
05D10 - Ramsey theory
11B75 - Combinatorial number theory
11F99 - None of the above but in this section
11N99 - None of the above but in this section
11B30 - Arithmetic combinatorics; higher degree uniformity
Additional resources :
https://www.cirm-math.fr/RepOrga/2228/Slides/talk_CAM2020_Stevens.pdf
CIRM (Publisher )
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Abstract :
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'.
Keywords : sum-product problem; Erdös-Szemerédi; convex sets; product set
MSC Codes : $max(\left | A+A \right |,\left | A+A \right |)\geq \left | A \right |^{\frac{4}{3}+\frac{2}{1167}-o^{(1)}}$ , 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'.
Keywords : sum-product problem; Erdös-Szemerédi; convex sets; product set
05D10 - Ramsey theory
11B75 - Combinatorial number theory
11F99 - None of the above but in this section
11N99 - None of the above but in this section
11B30 - Arithmetic combinatorics; higher degree uniformity
Additional resources :
https://www.cirm-math.fr/RepOrga/2228/Slides/talk_CAM2020_Stevens.pdf
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 29/09/2020 Conference Date : 07/09/2020 Subseries : Research talks arXiv category : Combinatorics ; Number Theory Mathematical Area(s) : Combinatorics ; Number Theory Format : MP4 (.mp4) - HD Video Time : 00:34:42 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2020-09-07_Stevens.mp4 
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Information on the Event
Event Title : Additive Combinatorics / Combinatoire additiveEvent Organizers : Balandraud, Eric ; Dousse, Jehanne ; Girard, Benjamin ; Schmid, Wolfgang ; Tringali, Salvatore Dates : 07/09/2020 - 11/09/2020 Event Year : 2020 Event URL : https://conferences.cirm-math.fr/2228.html
Citation Data
DOI : 10.24350/CIRM.V.19654203Cite this video as: Stevens, Sophie (2020). An update on the sum-product problem in $\mathbb{R}$. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19654203 URI : http://dx.doi.org/10.24350/CIRM.V.19654203 |
See Also
- [Multi angle] Spectra of ultra-discrete limits / Author of the conference Zuk, Andrzej.
- [Virtualconference] Zero-sum squares in bounded discrepancy {-1,1}-matrices / Author of the conference Montejano, Amanda.
- [Virtualconference] Higher order uniformity of the Möbius function / Author of the conference Matomäki, Kaisa.
- [Virtualconference] On the local distribution of the product of two shifted primes and application / Author of the conference Deshouillers, Jean-Marc.
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
