Abstract:
We show that every set S ⊆[N]d occupying ⋘ pκ residue classes for some real number 0 ≤κ < d and every prime p, must essentially lie in the solution set of a polynomial equation of degree ⋘(log N)C, for some constant C depending only on κ and d. This provides the first structural result for arbitrary κ < d and S. © 2014 Springer Basel.
Referencias:
- Breuillard, E., Green, B., Tao, T., The structure of approximate groups (2012) Institut Des Hautes Études Scientifiques, 116, pp. 115-221
- Croot, E., Lev, V., Open problems in additive combinatorics (2007) Additive Combinatorics, CRM Proc. Lecture Notes, 43, pp. 207-233. , Amer. Math. Soc., Providence
- Elsholtz, C., The distribution of sequences in residue classes (2002) Proceedings of the American Mathematical Society, 130 (8), pp. 2247-2250
- Green, B., On a variant of the large sieve, , (unpublished)
- Green, B., Harper, A., Inverse questions for the large, , sieve (unpublished)
- Green, B., Tao, T., Ziegler, T., An inverse theorem for the Gowers Us+1[N]-norm (2012) Annals of Mathematics, 176 (2), pp. 1231-1372
- Helfgott, H.A., Growth and generation in SL2(ℤ}/pℤ) (2008) Annals of Mathematics, 167 (2), pp. 601-623
- Helfgott, H.A., Venkatesh, A., How small must ill-distributed sets be? (2009) Analytic Number Theory. Essays in Honour of Klaus Roth, pp. 224-234. , Cambridge University Press, Cambridge
- Kowalski, E., The large sieve and its applications: arithmetic geometry, random walks and discrete groups (2008) Cambridge Tracts in Math., 175. , Cambridge University Press, Cambridge
- Tao, T., Vu, V., Inverse Littlewood-Offord theorems and the condition number of random matrices (2009) Annals of Mathematics, 169 (2), pp. 595-632
- Walsh, M., The inverse sieve problem in high dimensions (2012) Duke Mathematical Journal, 161 (10), pp. 2001-2022
Citas:
---------- APA ----------
(2014)
. The Algebraicity of Ill-Distributed Sets. Geometric and Functional Analysis, 24(3), 959-967.
http://dx.doi.org/10.1007/s00039-014-0286-3---------- CHICAGO ----------
Walsh, M.N.
"The Algebraicity of Ill-Distributed Sets"
. Geometric and Functional Analysis 24, no. 3
(2014) : 959-967.
http://dx.doi.org/10.1007/s00039-014-0286-3---------- MLA ----------
Walsh, M.N.
"The Algebraicity of Ill-Distributed Sets"
. Geometric and Functional Analysis, vol. 24, no. 3, 2014, pp. 959-967.
http://dx.doi.org/10.1007/s00039-014-0286-3---------- VANCOUVER ----------
Walsh, M.N. The Algebraicity of Ill-Distributed Sets. Geom. Funct. Anal. 2014;24(3):959-967.
http://dx.doi.org/10.1007/s00039-014-0286-3