Geodesic
On the Littlewood–Offord problem
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 21, Tome 431 (2014), pp. 72-81 Cet article a éte moissonné depuis la source Math-Net.Ru

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The paper deals with studying a connection of the Littlewood–Offord problem with estimating the concentration functions of some symmetric infinitely divisible distributions. Some multivariate generalizations of results of Arak (1980) are given. They show a connection of the concentration function of the sum with the arithmetic structure of supports of distributions of independent random vectors for arbitrary distributions of summands. 
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Yu. S. Eliseeva; A. Yu. Zaitsev. On the Littlewood–Offord problem. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 21, Tome 431 (2014), pp. 72-81. http://geodesic.mathdoc.fr/item/ZNSL_2014_431_a4/

[1] T. V. Arak, “O sblizhenii $n$-kratnykh svertok raspredelenii, imeyuschikh neotritsatelnuyu kharakteristicheskuyu funktsiyu, s soprovozhdayuschimi zakonami”, Teoriya veroyatn. i ee primen., 25:2 (1980), 225–246 | MR | Zbl

[2] T. V. Arak, “O skorosti skhodimosti v ravnomernoi predelnoi teoreme Kolmogorova. I”, Teoriya veroyatn. i ee primen., 26:2 (1981), 225–245 | MR | Zbl

[3] T. V. Arak, A. Yu. Zaitsev, Ravnomernye predelnye teoremy dlya summ nezavisimykh sluchainykh velichin, Tr. MIAN SSSR, 174, 1986, 217 pp. | MR | Zbl

[4] Yu. S. Eliseeva, “Mnogomernye otsenki funktsii kontsentratsii vzveshennykh summ nezavisimykh odinakovo raspredelennykh sluchainykh velichin”, Zap. nauchn. semin. POMI, 412, 2013, 121–137

[5] Yu. S. Eliseeva, F. Gëttse, A. Yu. Zaitsev, “Otsenki funktsii kontsentratsii v probleme Littlvuda–Offorda”, Zap. nauchn. semin. POMI, 420, 2013, 50–69

[6] Yu. S. Eliseeva, A. Yu. Zaitsev, “Otsenki funktsii kontsentratsii vzveshennykh summ nezavisimykh odinakovo raspredelennykh sluchainykh velichin”, Teoriya veroyatn. i ee primen., 57:4 (2012), 768–777 | DOI | Zbl

[7] C.-G. Esséen, “On the Kolmogorov–Rogozin inequality for the concentration function”, Z. Wahrscheinlichkeitstheorie Verw. Geb., 5 (1966), 210–216 | DOI | MR | Zbl

[8] C.-G. Esséen, “On the concentration function of a sum of independent random variables”, Z. Wahrscheinlichkeitstheorie Verw. Geb., 9 (1968), 290–308 | DOI | MR | Zbl

[9] O. Friedland, S. Sodin, “Bounds on the concentration function in terms of Diophantine approximation”, C. R. Math. Acad. Sci. Paris, 345 (2007), 513–518 | DOI | MR | Zbl

[10] G. Halász, “Estimates for the concentration function of combinatorial number theory and probability”, Periodica Mathematica Hungarica, 8 (1977), 197–211 | DOI | MR | Zbl

[11] V. Khengartner, R. Teodoresku, Funktsii kontsentratsii, Nauka, M., 1980 | MR

[12] J. E. Littlewood, A. C. Offord, “On the number of real roots of a random algebraic equation (III)”, Matem. sb., 12(54):3 (1943), 277–286 | MR | Zbl

[13] H. Nguyen, V. Vu, “Optimal inverse Littlewood–Offord theorems”, Adv. Math., 226 (2011), 5298–5319 | DOI | MR | Zbl

[14] H. Nguyen, V. Vu, Small probabilities, inverse theorems and applications, 2013, arXiv: 1301.0019

[15] V. V. Petrov, Summy nezavisimykh sluchainykh velichin, Nauka, M., 1972 | MR

[16] M. Rudelson, R. Vershynin, “The Littlewood–Offord problem and invertibility of random matrices”, Adv. Math., 218 (2008), 600–633 | DOI | MR | Zbl

[17] M. Rudelson, R. Vershynin, “The smallest singular value of a random rectangular matrix”, Comm. Pure Appl. Math., 62 (2009), 1707–1739 | DOI | MR | Zbl

[18] T. Tao, V. Vu, “Inverse Littlewood–Offord theorems and the condition number of random discrete matrices”, Ann. Math., 169 (2009), 595–632 | DOI | MR | Zbl

[19] T. Tao, V. Vu, “From the Littlewood–Offord problem to the circular law: universality of the spectral distribution of random matrices”, Bull. Amer. Math. Soc., 46 (2009), 377–396 | DOI | MR | Zbl

[20] R. Vershynin, “Invertibility of symmetric random matrices”, Random Structures Algorithms, 44 (2014), 135–182 | DOI | MR | Zbl

[21] A. Yu. Zaitsev, “K mnogomernomu obobscheniyu metoda treugolnykh funktsii”, Zap. nauchn. semin. LOMI, 158, 1987, 81–104 | MR | Zbl