Abstract
In this short survey, we discuss the notion of anti-concentration and describe various ideas used to obtain anti-concentration inequalities, together with several open questions.
In memory of Jirka Matoušek
V. Vu is supported by NSF grant DMS-1307797 and AFORS grant FA9550-12-1-0083.
This is a preview of subscription content, log in via an institution to check access.
References
A. Carbery, J. Wright, Distributional and L q norm inequalities for polynomials over convex bodies in R n. Math. Res. Lett. 8(3), 233–248 (2001)
K. Costello, Bilinear and quadratic variants on the Littlewood-Offord problem. Israel J. Math. 194.1, 359–394 (2013)
K. Costello, T. Tao, V. Vu, Random symmetric matrices are almost surely nonsingular. Duke Math. J. 135.2, 395–413 (2006)
I. Diakonikolas, R. Servedio, L. Tan, A. Wan, A regularity lemma, and low-weight approximators, for low-degree polynomial threshold functions, in 2010 IEEE 25th Annual Conference on Computational Complexity (CCC) (2010), pp. 211–222
P. Erdös, On a lemma of Littlewood and Offord. Bull. Am. Math. Soc. 51.12, 898–902 (1945)
C. Esséen, On the Kolmogorov-Rogozin inequality for the concentration function. Z. Wahrsch. Verw. Gebiete 5, 210–216 (1966)
G. Halász, Estimates for the concentration function of combinatorial number theory and probability. Period. Math. Hungar. 8(3–4), 197–211 (1977)
D. Kane, The Correct Exponent for the Gotsman-Linial Conjecture, in 2013 IEEE 28th Conference on Computational Complexity (CCC) (IEEE, 2013), pp. 56–64
D. Kane, A pseudorandom generator for polynomial threshold functions of gaussian with subpolynomial seed length, in 2014 IEEE 29th Conference on Computational Complexity (CCC) (IEEE, 2014), pp. 217–228
M. Ledoux, The Concentration of Measure Phenomenon. Volume 89 of Mathematical Surveys and Mono-graphs. (American Mathematical Society, 2001)
J.W. Lindeberg, Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung. Math. Z. 15, 211–225 (1922)
J.E. Littlewood, A.C. Offord, On the number of real roots of a random algebraic equation. III. Rec. Math. [Mat. Sbornik] N.S. 54, 277–286 (1943)
R. Meka, O. Nguyen, V. Vu, Anti-concentration for polynomials of independent random variables. Theory of Comput. 12(11), 1–17 (2016)
E. Mossel, R. O’Donnell, K. Oleszkiewicz, Noise stability of functions with low influences: invariance and optimality. Ann. Math. 171(1), 295–341 (2010)
H. Nguyen, Inverse Littlewood-Offord problems and the singularity of random symmetric matrices. Duke Math. J. 161(4), 545–586 (2012)
H. Nguyen, V. Vu, Small Ball Probability, Inverse Theorems, and Applications, Erdös Centennial (Springer, Berlin/Heidelberg, 2013), pp. 409–463
A. Razborov, E. Viola, Real advantage. ACM Trans. Comput. Theory (TOCT) 5(4), 17 (2013)
M. Rudelson, R. Vershynin, Non-asymptotic theory of random matrices: extremal singular values, in Proceeding of the International Congress of Mathematicians, vol. III (Hindustan Book Agency, New Delhi, 2010), pp. 1576–1602
T. Tao, V. Vu, From the Littlewood-Offord problem to the circular law: universality of the spectral distribution of random matrices. Bull. Am. Math. Soc. (N.S.) 46(3), 377–396 (2009)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International publishing AG
About this chapter
Cite this chapter
Vu, V. (2017). Anti-concentration Inequalities for Polynomials. In: Loebl, M., Nešetřil, J., Thomas, R. (eds) A Journey Through Discrete Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-44479-6_32
Download citation
DOI: https://doi.org/10.1007/978-3-319-44479-6_32
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-44478-9
Online ISBN: 978-3-319-44479-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)