Geodesic
On sum sets of sets having small product set
Informatics and Automation, Modern problems of mathematics, mechanics, and mathematical physics, Tome 290 (2015), pp. 304-316.

Voir la notice de l'article provenant de la source Math-Net.Ru

We improve the sum–product result of Solymosi in $\mathbb R$; namely, we prove that $\max \{|A+A|,|AA|\}\gg |A|^{4/3+c}$, where $c>0$ is an absolute constant. New lower bounds for sums of sets with small product set are found. Previous results are improved effectively for sets $A\subset \mathbb R$ with $|AA| \le |A|^{4/3}$. 
@article{TRSPY_2015_290_a24,
     author = {S. V. Konyagin and I. D. Shkredov},
     title = {On sum sets of sets having small product set},
     journal = {Informatics and Automation},
     pages = {304--316},
     publisher = {mathdoc},
     volume = {290},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2015_290_a24/}
}
TY  - JOUR
AU  - S. V. Konyagin
AU  - I. D. Shkredov
TI  - On sum sets of sets having small product set
JO  - Informatics and Automation
PY  - 2015
SP  - 304
EP  - 316
VL  - 290
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2015_290_a24/
LA  - ru
ID  - TRSPY_2015_290_a24
ER  - 
%0 Journal Article
%A S. V. Konyagin
%A I. D. Shkredov
%T On sum sets of sets having small product set
%J Informatics and Automation
%D 2015
%P 304-316
%V 290
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TRSPY_2015_290_a24/
%G ru
%F TRSPY_2015_290_a24
S. V. Konyagin; I. D. Shkredov. On sum sets of sets having small product set. Informatics and Automation, Modern problems of mathematics, mechanics, and mathematical physics, Tome 290 (2015), pp. 304-316. http://geodesic.mathdoc.fr/item/TRSPY_2015_290_a24/

[1] Bourgain J., Garaev M.Z., “On a variant of sum–product estimates and explicit exponential sum bounds in prime fields”, Math. Proc. Cambridge Philos. Soc., 146:1 (2009), 1–21 | DOI | MR | Zbl

[2] Elekes G., Ruzsa I., “Few sums, many products”, Stud. sci. math. Hung., 40:3 (2003), 301–308 | MR | Zbl

[3] Erdős P., Szemerédi E., “On sums and products of integers”, Studies in pure mathematics, To the memory of Paul Turán, Birkhäuser, Basel, 1983, 213–218 | MR

[4] Katz N.H., Koester P., “On additive doubling and energy”, SIAM J. Discrete Math., 24 (2010), 1684–1693 | DOI | MR | Zbl

[5] Raz O.E., Roche-Newton O., Sharir M., “Sets with few distinct distances do not have heavy lines”, Discrete Math., 338:8 (2015), 1484–1492, arXiv: 1410.1654v1 [math.CO] | DOI | MR | Zbl

[6] Schoen T., “New bounds in Balog–Szemerédi–Gowers theorem”, Combinatorica, 2014 | DOI

[7] Schoen T., Shkredov I.D., “Higher moments of convolutions”, J. Number Theory, 133:5 (2013), 1693–1737 | DOI | MR | Zbl

[8] Shkredov I.D., “Some new inequalities in additive combinatorics”, Moscow J. Comb. Number Theory, 3 (2013), 189–239 | MR | Zbl

[9] Shkredov I.D., “O summakh mnozhestv Semeredi–Trottera”, Tr. MIAN, 289 (2015), 318–327, arXiv: 1410.5662v1 [math.CO] | MR

[10] Solymosi J., “Bounding multiplicative energy by the sumset”, Adv. Math., 222:2 (2009), 402–408 | DOI | MR | Zbl

[11] Tao T., Vu V.H., Additive combinatorics, Cambridge Univ. Press, Cambridge, 2006 | MR | Zbl