Small ball probability, inverse theorems, and applications

Bolyai Society Mathematical Studies

Volumn 25, Issue , 2013, Pages 409-463

  Nguyen, Hoi H ^a   Vu, Van H ^a  

^a YALE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

INVERSE PROBLEMS;

ITS APPLICATIONS; MAXIMUM PROBABILITY; MULTI-SET;

PROBABILITY;

EID: 84974786849     PISSN: 12174696     EISSN: None     Source Type: Book Series    
DOI: 10.1007/978-3-642-39286-3_16     Document Type: Chapter

References (70)

1. B. Bollobás, Random Graphs, Academic Press, New York.
2. C. Bordenave and D. Chafai, Around the circular law, Probab. Surveys 9 (2012), 1-89.
3. J. Bourgain, V. Vu and P. M. Wood, On the singularity probability of discrete random matrices, Journal of Functional Analysis 258 (2010), no.2, 559-603.
4. A. T. Bharucha-Reid and M. Sambandham, Random polynomials, Academic Press, Orlando, 1986.
5. E. Breuillard, B. Green and Terence Tao, The structure of approximate groups, to appear in Pub. IHES, http://arxiv.org/abs/1110.5008.
6. D. Conlon, J. Fox and B. Sudakov, Essays in extremal combinatorics, submitted, http://arxiv.org/abs/1212.1300.
7. K. Costello, Bilinear and quadratic variants on the Littlewood-Offord problem, to appear in Israel of Mathematics, http://arxiv.org/abs/0902.1538.
8. K. Costello, T. Tao and V. Vu, Random symmetric matrices are almost surely non-singular, Duke Math. J. 135 (2006), 395-413.
9. A. Edelman, Eigenvalues and condition numbers of random matrices, SIAM J. Matrix Anal. Appl. 9 (1988), no. 4, 543-560.
10. P. Erdős, On a lemma of Littlewood and Offord, Bull. Amer. Math. Soc. 51 (1945), 898-902.
11. P. Erdős and L. Moser, Elementary Problems and Solutions, Amer. Math. Monthly, 54 (1947), no. 4, 229-230.
12. C. G. Esséen, On the Kolmogorov-Rogozin inequality for the concentration function, Z. Wahrsch. Verw. Gebiete 5 (1966), 210-216.
13. W. Feller, An introduction to probability and its applications, Wiley series in probability and mathematical statistics.
14. P. Frankl and Z. Füredi, Solution of the Littlewood-Offord problem in high dimensions, Ann. of Math. (2) 128 (1988), no. 2, 259-270.
15. G. Freiman, Foundations of a Structural Theory of Set Addition, Translations of Mathematical Monographs 37, Amer. Math. Soc, Providence, RI, USA, 1973.
16. O. Friedland and S. Sodin, Bounds on the concentration function in terms of Diophantine approximation, C. R. Math. Acad. Sci. Paris 345 (2007), no. 9, 513-518.
17. H. Goldstine and J. von Neumann, Numerical inverting of matrices of high order, Bull. Amer. Math. Soc. 53 (1947), 1021-1099.
18. F. Götze and A. Tikhomirov, The circular law for random matrices, Ann. Probab. 38 (2010), no. 4, 1444-1491.
19. J. Griggs, The Littlewood-Offord problem: tightest packing and an M-part Sperner theorem, Europ. J. Combin. 1 (1980), 225-234.
20. D. S. Gunderson, V. Rödl and A. Sidorenko, Extremal problems for sets forming Boolean algebras and complete partite hypergraphs, J. Combin. Theory Ser. A 88 (1999), 342-367.
21. G. Halász, Estimates for the concentration function of combinatorial number theory and probability, Period. Math. Hungar. 8 (1977), no. 3-4, 197-211.
22. D. Hilbert, Über die Irreduzibilität ganzer rationaler Funktionen mit ganzzahligen Koeffizienten, J. Reine Angew. Math. 110 (1892), 104-129.
23. J. Kahn, J. Komlós and E. Szemerédi, On the probability that a random ±1 matrix is singular, J. Amer. Math. Soc. 8 (1995), 223-240.
24. G. Katona, On a conjecture of Erdős and a stronger form of Sperner’s theorem. Studia Sci. Math. Hungar 1 (1966), 59-63.
25. D. Kleitman, On a lemma of Littlewood and Offord on the distributions of linear combinations of vectors, Advances in Math. 5 1970 155-157 (1970).
26. D. Kleitman, Some new results on the Littlewood-Offord problem, J. Combinatorial Theory Ser. A 20 (1976), no. 1, 89-113.
27. D. Kleitman, On a lemma of Littlewood and Offord on the distribution of certain sums, Math. Z. 90 1965 251-259.
28. J. Komlós, On the determinant of (0; 1) matrices, Studia Sci. Math. Hungar. 2 (1967), 7-22.
29. J. Komlós, On the determinant of random matrices, Studia Sci. Math. Hungar. 3 (1968), 387-399.
30. A. Kolmogorov, Two uniform limit theorems for sums of independent random variables, Theor. Probab. Appl. 1 (1956), 384-394.
31. A. Kolmogorov, Sur les propriétés des fonctions de concentrations de M. P. Lévy, Ann. Inst. H. Poincaré 16 (1958), 27-34.
32. G. Kozma and O. Zeitouni, On common roots of random Bernoulli polynomials, to appear in Int. Math. Res. Not., http://arxiv.org/abs/1109.2316.
33. J. E. Littlewood and A. C. Offord, On the number of real roots of a random algebraic equation. III. Rec. Math. Mat. Sbornik N.S. 12, (1943). 277-286.
34. S. Muroga, I. Toda, and S. Takasu, Theory of majority decision elements, J. Franklin Inst., 271, 376-418, 1961.
35. S. Muroga, Threshold logic and its applications, Wiley-Interscience, New York, 1971.
36. H. Nguyen, Inverse Littlewood-Offord problems and the singularity of random symmetric matrices, Duke Mathematics Journal Vol. 161, 4 (2012), 545-586.
37. H. Nguyen, A new approach to an old problem of Erdős and Moser, Journal of Combinatorial Theory, Series A 119 (2012) 977-993.
38. H. Nguyen, Singularity of random combinatorial matrices, to appear in SIAM J. Discrete Mathematics, http://arxiv.org/abs/1112.0753.
39. H. Nguyen and V. Vu, Optimal Littlewood-Offord theorems, Advances in Math., Vol. 226 6 (2011), 5298-5319.
40. A. Pajor and L. Pastur, On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution, Studia Math. 195 (2009), no. 1, 11-29.
41. R. A. Proctor, Solution of two difficult combinatorial problems with linear algebra, Amer. Math. Monthly 89 (1982), no. 10, 721-734.
42. A. Razborov and E. Viola, Real Advantage, submitted, http://eccc.hpi-web.de/report/2012/134/.
43. B. A. Rogozin, An estimate for concentration functions, Theor. Probab. Appl. 6 (1961), 94-97.
44. M. Rudelson, Invertibility of random matrices: Norm of the inverse, Annals of Mathematics, 168 (2008), no. 2, 575-600.
45. M. Rudelson and R. Vershynin, The Littlewood-Offord Problem and invertibility of random matrices, Advances in Mathematics 218 (2008), 600-633.
46. M. Rudelson and R. Vershynin, Smallest singular value of a random rectangular matrix, Communications on Pure and Applied Mathematics 62 (2009), 1707-1739.
47. M. Rudelson and R. Vershynin, Non-asymptotic theory of random matrices: extreme singular values, Proceedings of the International Congress of Mathematicians. Volume III, 1576-1602, Hindustan Book Agency, New Delhi, 2010.
48. A. Sali, Strong from of an M-part Sperner theorem, European J. Combinatorics 4 (1983), 179-183.
49. A. Sali, A Sperner type theorem, Order 2 (1985), 13-127.
50. A. Sárközy and E. Szemerédi, Über ein Problem von Erdős und Moser, Acta Arithmetica 11 (1965), 205-208.
51. A. A. Sherstov, Communication lower bounds using dual polynomials, Bulletin of the EATCS, 95, 59-93, 2008.
52. D. A. Spielman and S. H. Teng, Smoothed analysis of algorithms, Proceedings of the International Congress of Mathematicians, Vol. 1, 597-606, Higher Ed. Press, Beijing, 2002.
53. D. A. Spielman and S. H. Teng, Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time, J. ACM 51 (2004), no. 3, 385-463.
54. R. Stanley, Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Algebraic Discrete Methods 1 (1980), no. 2, 168-184.
55. E. Szemerédi, On sets of integers containing no four elements in arithmetic progression, Acta Math. Acad. Sci. Hungar. 20 (1969), 199-245.
56. T. Tao and V. Vu, On random ±1 matrices: singularity and determinant, Random Structures Algorithms 28 (2006), 1-23.
57. T. Tao and V. Vu, On the singularity probability of random Bernoulli matrices, Journal of the A. M. S 20 (2007), 603-673.
58. T. Tao and V. Vu, Random matrices: The Circular Law, Communication in Contemporary Mathematics 10 (2008), 261-307.
59. T. Tao and V. Vu, From the Littlewood-Offord problem to the circular law: universality of the spectral distribution of random matrices, Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 3, 377-396.
60. T. Tao and V. Vu, Inverse Littlewood-Offord theorems and the condition number of random matrices, Annals of Mathematics (2) 169 (2009), no. 2, 595-632.
61. T. Tao and V. Vu, On the permanent of random Bernoulli matrices, Adv. Math. 220 (2009), 657-669.
62. T. Tao and V. Vu, A sharp inverse Littlewood-Offord theorem, Random Structures Algorithms 37 (2010), no. 4, 525-539.
63. T. Tao and V. Vu, Smooth analysis of the condition number and the least singular value, Mathematics of Computation 79 (2010), 2333-2352.
64. T. Tao and V. Vu, Random matrices: the distribution of the smallest singular values, Geom. Funct. Anal. 20 (2010), no. 1, 260-297.
65. T. Tao and V. Vu, Random matrices: universality of ESDs and the circular law, Ann. Probab. 38 (2010), no. 5 p. 2023-2065, with an appendix by M. Krishnapur.
66. T. Tao and V. Vu, The Littlewood-Offord problem in high dimensions and a conjecture of Frankl and Füredi, Combinatorica 32 (2012), no. 3, 363-372.
67. T. Tao and V. Vu, Additive Combinatorics, Cambridge Univ. Press, 2006.
68. L. G. Valiant, Graph-theoretic arguments in low-level complexity, In Proceedings of the 6th MFCS, Lecture Notes in Computer Science, 53, p. 162-176, New York/Berlin, 1977, Springer-Verlag.
69. R. Vershynin, Invertibility of symmetric random matrices, to appear in Random Structures and Algorithms, http://arxiv.org/abs/1102.0300.
70. E. Viola, On the power of small depth computation, Foundations and Trends in Theoretical Computer Science, 5(1), 1-72, 2009.