Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-05T04:18:15.444Z Has data issue: false hasContentIssue false
  • English
  • Français

On higher energy decompositions and the sum–product phenomenon

Published online by Cambridge University Press:  03 July 2018

GEORGE SHAKAN
GEORGE SHAKAN*
Affiliation:
Department of Mathematics, University of Illinois, Urbana–Champaign e-mail: shakan2@illinois.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Let A ⊂ ℝ be finite. We quantitatively improve the Balog–Wooley decomposition, that is A can be partitioned into sets B and C such that

$ \begin{equation*} \max\{E^+(B) , E^{\times}(C)\} \lesssim |A|^{3 - 7/26}, \ \ \max \{E^+(B,A) , E^{\times}(C, A) \}\lesssim |A|^{3 - 1/4}. \end{equation*} $
We use similar decompositions to improve upon various sum–product estimates. For instance, we show
$ \begin{equation*} |A+A| + |A A| \gtrsim |A|^{4/3 + 5/5277}. \end{equation*} $

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 167 , Issue 3 , November 2019 , pp. 599 - 617
Copyright
Copyright © Cambridge Philosophical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[BaWo] Balog, A. and Wooley, T.. A low–energy decomposition theorem. Q. J. Math. 68.1 (2017), 207–226.Google Scholar
[BaRo] Balog, A. and Roche–Newton, O.. New sum–product estimates for real and complex numbers, Comput. Geom. 53 (2015), no. 4, 825–846.Google Scholar
[BoCh] Bourgain, J. and Chang, M–C.. On the size of k–fold sum and product sets of integers. J. Amer. Math. Soc. 17 (2) (2004), 473–497.Google Scholar
[Ch] Chang, M–C.. The Erdős–Szemerédi problem on sum set and product set. Ann. of Math. (2) 157 (3) (2003), 939–957.Google Scholar
[CrHa] Croot, E. and Hart, D.. h–fold sums from a set with few products. SIAM J. Discrete Math. 24 (2010), 505–519.Google Scholar
[El] Elekes, G.. On the number of sums and products. Acta Arith. 81 (1997), 365–367.Google Scholar
[ElRu] Elekes, G. and Ruzsa, I.. Few sums, many products. Studia Sci. Math. Hungar. 40 (3) (2003), 301–308.Google Scholar
[ErSz] Erdős, P. and Szemerédi, E.. On sums and products of integers. Studies in Pure Math. (Birkhauser, Basel, 1983), 213–218.Google Scholar
[Fo] Ford, K.. Sums and products from a finite set of real numbers. Ramanujan J. 2 (1998), 59–66.Google Scholar
[Ha] Hanson, B.. Estimates for character sums with various convolutions. Preprint (2015) arXiv:1509.04354.Google Scholar
[KaKo] Katz, N. H. and Koester, P.. On additive doubling and energy. SIAM Journal on Discrete Mathematics 24 (2010), 1684–1693.Google Scholar
[IRR] Iosevich, A., Roche–Newton, O. and Rudnev, M.. On discrete values of bilinear forms. arXiv: 1512.02670 (2015).Google Scholar
[KoRu] Konyagin, S. and Rudnev, M.. On new sum–product type estimates. SIAM J. Discrete Math. 27 (2) (2013), 973–990.Google Scholar
[KoSh1] Konyagin, S. and Shkredov, I.. On sum sets of sets, having small product set. Proc. Steklov Inst. Math. 290 (2015), 288–299.Google Scholar
[KoSh2] Konyagin, S. and Shkredov, I.. New results on sum–products in R. Trans. Steklov Math. Inst. 294 (2016), 87–98.Google Scholar
[LiRo] Li, L. and Roche–Newton, O.. Convexity and a sum–product estimate. Acta Arith. 156 (2012), 247–255.Google Scholar
[LiSh] Li, L. and Shen, J.. A sum–division estimate of reals. Proc. Amer. Math. Soc. 138 (1) (2010), 101–104.Google Scholar
[MRS1] Murphy, B., Roche–Newton, O. and Shkredov, I.. Variations of the sum–product problem, arXiv: 1312.6438 (2014).Google Scholar
[MRS2] Murphy, B., Roche–Newton, O. and Shkredov, I.. Variations of the sum–product problem II, arXiv: 1703.09549 (2017).Google Scholar
[MRSS] Murphy, B., Rudnev, M., Shkredov, I. and Shteinikov, Y.. On the few products, many sums problem. arXiv: 1712.0041v1 (2017).Google Scholar
[Na] Nathanson, M.. On sums and products of integers. Proc. Amer. Math. Soc. 125 (1997), 9–16.Google Scholar
[RRSS] Roche–Newton, O., Ruzsa, I., Shen, C. and Shkredov, I.. On the size of the set $AA+A$. arXiv: 1801.1043v1 (2018).Google Scholar
[RSS] Rudnev, M., Shkredov, I. and Stevens, S.. On an energy variant of the sum–product conjecture. arXiv: 1607.05053 (2016).Google Scholar
[ScSh] Schoen, T. and Shkredov, I.. On sumsets of convex sets. Comb. Probab. Comput. 20 (2011), 793–798.Google Scholar
[Sha] Shakan, G.. Konyagin–Shkredov Clustering. Blog Post. Mar. 10 (2018). https://gshakan.wordpress.com/konyagin-shkredov-clustering/.Google Scholar
[She] Sheffer, A.. Konyagain–Shkredov sum–product bound. Blog Post. July 9 (2016). https://adamsheffer.files.wordpress.com/2016/07/ks-sp.pdf.Google Scholar
[Sh1] Shkredov, I.. On sums of Szemerédi–Trotter sets. Proc. Steklov Inst. Math. 289 (1) (2015), 300–309.Google Scholar
[Sh2] Shkredov, I. D.. Some remarks on the Balog–Wooley decomposition theorem and quantities D +, D ×. Proc. Steklov Inst. Math. Accepted, arXiv:1605.00266v1 (2016).Google Scholar
[Sh3] Shkredov, I.. Some new results on higher energies. Trans. MMS 74 (1) (2013), 35–73.Google Scholar
[Sh4] Shkredov, I.. On a question of A. Balog. arXiv: 1501:07498 (2015).Google Scholar
[So1] Solymosi, J.. On the number of sums and products. Bull. Lond. Math. Soc. 37 (4) (2005), 491–494.Google Scholar
[So2] Solymosi, J.. Bounding multiplicative energy by the sumset. Adv. Math. 222 (2) (2009), 402–408.Google Scholar
[TV] Tao, T. and Vu, V.. Additive Combinatorics (Cambridge University Press, 2006).Google Scholar