Geodesic
Additive Properties of Slowly Increasing Convex Sets
Matematičeskie zametki, Tome 108 (2020) no. 6, pp. 851-867.

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

We obtain new estimates for the distribution of convolutions of the set of values of a convex function at integer points under additional conditions on the higher derivatives of the function. New estimates for additive energy and for the dimension of sumsets and sets of differences of such sets arise as natural consequences.
Keywords: additive combinatorics, additive energy, sumsets, convex sets, convex sequences. 
@article{MZM_2020_108_6_a3,
     author = {K. I. Olmezov},
     title = {Additive {Properties} of {Slowly} {Increasing} {Convex} {Sets}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {851--867},
     publisher = {mathdoc},
     volume = {108},
     number = {6},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2020_108_6_a3/}
}
TY  - JOUR
AU  - K. I. Olmezov
TI  - Additive Properties of Slowly Increasing Convex Sets
JO  - Matematičeskie zametki
PY  - 2020
SP  - 851
EP  - 867
VL  - 108
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2020_108_6_a3/
LA  - ru
ID  - MZM_2020_108_6_a3
ER  - 
%0 Journal Article
%A K. I. Olmezov
%T Additive Properties of Slowly Increasing Convex Sets
%J Matematičeskie zametki
%D 2020
%P 851-867
%V 108
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2020_108_6_a3/
%G ru
%F MZM_2020_108_6_a3
K. I. Olmezov. Additive Properties of Slowly Increasing Convex Sets. Matematičeskie zametki, Tome 108 (2020) no. 6, pp. 851-867. http://geodesic.mathdoc.fr/item/MZM_2020_108_6_a3/

[1] N. Hegyvári, “On consecutive sums in sequences”, Acta Math. Hungar., 48:1-2 (1986), 193–200 | DOI | MR | Zbl

[2] T. Tao, V. Vu, Additive Combinatorics, Cambridge University Press, Cambridge, 2006 | MR | Zbl

[3] T. Schoen, I. D. Shkredov, “On sumsets of convex sets”, Combin. Probab. Comput., 20:5 (2011), 793–798 | DOI | MR | Zbl

[4] M. Rudnev, S. Stevens, An Update on the Sum-Product Problem, 2020, arXiv: 2005.11145

[5] I. D. Shkredov, “Neskolko novykh rezultatov o starshikh energiyakh”, Tr. MMO, 74, no. 1, MTsNMO, M., 2013, 35–73 | MR | Zbl

[6] S. V. Konyagin, “Ob otsenke $L_1$-normy odnoi eksponentsialnoi summy”, Teoriya priblizhenii funktsii i operatorov, Tezisy dokladov Mezhdunarodnoi konferentsii, posvyaschennoi 80-letiyu so dnya rozhdeniya S. B. Stechkina, Ekaterinburg, 2000, 88–89

[7] M. Z. Garaev, “O nizhnikh otsenkakh $L_1$-normy nekotorykh eksponentsialnykh summ”, Matem. zametki, 68:6 (2000), 842–850 | DOI | MR | Zbl

[8] S. V. Bochkarev, “Multiplikativnye neravenstva dlya $L_1$-normy, primeneniya v analize i teorii chisel”, Funktsionalnye prostranstva, teoriya priblizhenii, nelineinyi analiz, Tr. MIAN, 255, Nauka, MAIK «Nauka/Interperiodika», M., 2006, 55–70 | MR

[9] A. Iosevich, S. Konyagin, M. Rudnev, V. Ten, “On combinatorial complexity of convex sequences”, Discrete Comput. Geom., 35 (2006), 143–158 | DOI | MR | Zbl

[10] I. D. Shkredov, “Ob asimptoticheskikh formulakh v nekotorykh voprosakh teorii summ proizvedenii”, Tr. MMO, 79, no. 2, MTsNMO, M., 2018, 271–334 | MR