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Volumn 179, Issue 3, 2010, Pages 559-644

Affine linear sieve, expanders, and sum-product

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EID: 76549128679     PISSN: 00209910     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00222-009-0225-3     Document Type: Article
Times cited : (142)

References (64)
  • 2
    • 0001577658 scopus 로고
    • Arithmetic subgroups of algebraic groups
    • Borel, A., Harish-Chandra: Arithmetic subgroups of algebraic groups. Ann. Math. 75, 485-535 (1962).
    • (1962) Ann. Math. , vol.75 , pp. 485-535
    • Borel, A.1    Harish-Chandra2
  • 4
    • 18144429826 scopus 로고    scopus 로고
    • Mordell's exponential sum estimate revisited
    • Bourgain, J.: Mordell's exponential sum estimate revisited. J. Am. Math. Soc. 18, 477-499 (2005).
    • (2005) J. Am. Math. Soc. , vol.18 , pp. 477-499
    • Bourgain, J.1
  • 5
    • 33644955240 scopus 로고    scopus 로고
    • *, q arbitrary
    • *, q arbitrary. J. Anal. 97, 317-355 (2005).
    • (2005) J. Anal. , vol.97 , pp. 317-355
    • Bourgain, J.1
  • 6
    • 33745439041 scopus 로고    scopus 로고
    • *, where q is composite with few prime factors
    • *, where q is composite with few prime factors. Geom. Funct. Anal. 16, 327-366 (2006).
    • (2006) Geom. Funct. Anal. , vol.16 , pp. 327-366
    • Bourgain, J.1    Chang, M.-C.2
  • 9
    • 33646375214 scopus 로고    scopus 로고
    • Estimate for the number of sums and products and for exponential sums in fields of prime order
    • Bourgain, J., Glibichuk, A., Konyagin, S.: Estimate for the number of sums and products and for exponential sums in fields of prime order. J. Lond. Math. Soc. 73, 380-398 (2006).
    • (2006) J. Lond. Math. Soc. , vol.73 , pp. 380-398
    • Bourgain, J.1    Glibichuk, A.2    Konyagin, S.3
  • 10
    • 2442427378 scopus 로고    scopus 로고
    • A sum-product estimate in finite fields and applications
    • Bourgain, J., Katz, N., Tao, T.: A sum-product estimate in finite fields and applications. Geom. Funct. Anal. 14, 27-57 (2004).
    • (2004) Geom. Funct. Anal. , vol.14 , pp. 27-57
    • Bourgain, J.1    Katz, N.2    Tao, T.3
  • 11
    • 0038850876 scopus 로고
    • Le crible d'Eratosthène et le théoreme de Goldbach
    • Brun, V.: Le crible d'Eratosthène et le théoreme de Goldbach. Skr. Nor. Vidensk. Akad. Kristiania I 3, 1-36 (1920).
    • (1920) Skr. Nor. Vidensk. Akad. Kristiania I , vol.3 , pp. 1-36
    • Brun, V.1
  • 14
    • 0011690991 scopus 로고
    • Opredelenie plotnosti sovokupnosti prostykh chisel, prinadlezhashchikh zadannomu klassu podstanovok
    • Chebotarev, N. G.: Opredelenie plotnosti sovokupnosti prostykh chisel, prinadlezhashchikh zadannomu klassu podstanovok. Izv. Ross. Akad. Nauk. 17, 205-250 (1923).
    • (1923) Izv. Ross. Akad. Nauk. , vol.17 , pp. 205-250
    • Chebotarev, N.G.1
  • 15
    • 0037933477 scopus 로고    scopus 로고
    • Demonstration de la conjecture τ
    • Clozel, L.: Demonstration de la conjecture τ. Invent. Math. 151, 297-328 (2003).
    • (2003) Invent. Math. , vol.151 , pp. 297-328
    • Clozel, L.1
  • 16
    • 33746811131 scopus 로고    scopus 로고
    • Some applications of sieves of dimension exceeding 1
    • Cambridge: Cambridge University Press
    • Diamond, H., Halberstam, H.: Some applications of sieves of dimension exceeding 1. In: London Math. Soc. Lecture Note Ser., vol. 237, pp. 101-107. Cambridge University Press, Cambridge (1997).
    • (1997) London Math. Soc. Lecture Note Ser. , vol.237 , pp. 101-107
    • Diamond, H.1    Halberstam, H.2
  • 17
    • 76549106445 scopus 로고
    • Démonstration d'un théorème sur la progression arithmètique
    • Lejeune Dirichlet, G.: Démonstration d'un théorème sur la progression arithmètique. Abh. Preuss. Akad. Wiss. 45-71 (1837).
    • (1837) Abh. Preuss. Akad. Wiss , pp. 45-71
    • Lejeune Dirichlet, G.1
  • 19
    • 0001181799 scopus 로고
    • On Picard groups of algebraic fibre spaces
    • Fossum, R., Iversen, B.: On Picard groups of algebraic fibre spaces. J. Pure Appl. Algebra 3, 269-280 (1973).
    • (1973) J. Pure Appl. Algebra , vol.3 , pp. 269-280
    • Fossum, R.1    Iversen, B.2
  • 20
    • 61349166437 scopus 로고    scopus 로고
    • Hyperbolic prime number theorem
    • Friedlander, J., Iwaniec, H.: Hyperbolic prime number theorem. Acta Math. 202, 1-19 (2009).
    • (2009) Acta Math. , vol.202 , pp. 1-19
    • Friedlander, J.1    Iwaniec, H.2
  • 21
    • 76549099230 scopus 로고    scopus 로고
    • Friedlander, J., Iwaniec, H.: In preparation
    • Friedlander, J., Iwaniec, H.: In preparation.
  • 23
    • 76549133063 scopus 로고    scopus 로고
    • Fuchs, E.: Ph. D. thesis, in preparation
    • Fuchs, E.: Ph. D. thesis, in preparation.
  • 28
    • 28044467190 scopus 로고
    • Geneology of Pythagorean Triads
    • Hall, A.: Geneology of Pythagorean Triads. Math. Gazette 54(390), 377-379 (1970).
    • (1970) Math. Gazette , vol.54 , Issue.390 , pp. 377-379
    • Hall, A.1
  • 29
    • 33646862107 scopus 로고
    • Some problems of 'Partitio Numerorum': III. On the expression of a number as a sum of primes
    • Hardy, G. H., Littlewood, J. E.: Some problems of 'Partitio Numerorum': III. On the expression of a number as a sum of primes. Acta Math. 44, 1-70 (1922).
    • (1922) Acta Math. , vol.44 , pp. 1-70
    • Hardy, G.H.1    Littlewood, J.E.2
  • 30
    • 44449169647 scopus 로고    scopus 로고
    • 2(ℤ/pℤ)
    • 2(ℤ/pℤ). Ann. Math. 167, 601-623 (2008).
    • (2008) Ann. Math. , vol.167 , pp. 601-623
    • Helfgott, H.1
  • 32
    • 0040916920 scopus 로고
    • Primes represented by quadratic polynomials in two variables
    • Iwaniec, H.: Primes represented by quadratic polynomials in two variables. Acta Arith. 24, 435-459 (1973/74).
    • (1973) Acta Arith. , vol.24 , pp. 435-459
    • Iwaniec, H.1
  • 34
    • 0346530165 scopus 로고
    • Diophantine representation of the set of prime numbers
    • Jones, J., Sato, D., Wada, H., Wiens, D.: Diophantine representation of the set of prime numbers. Am. Math. Monthly 83, 449-464 (1976).
    • (1976) Am. Math. Monthly , vol.83 , pp. 449-464
    • Jones, J.1    Sato, D.2    Wada, H.3    Wiens, D.4
  • 35
    • 33847574798 scopus 로고
    • Symmetric random walks on groups
    • Kesten, H.: Symmetric random walks on groups. Trans. Am. Math. Soc. 92, 336-354 (1959).
    • (1959) Trans. Am. Math. Soc. , vol.92 , pp. 336-354
    • Kesten, H.1
  • 36
    • 0001005479 scopus 로고
    • Abschätzungen asymptotischen Dichte von Summenmengen
    • Kneser, M.: Abschätzungen asymptotischen Dichte von Summenmengen. Math. Z. 58, 459-484 (1953).
    • (1953) Math. Z. , vol.58 , pp. 459-484
    • Kneser, M.1
  • 39
    • 0012783530 scopus 로고    scopus 로고
    • 3rd edn, New York: Springer
    • Lang, S.: Algebra, 3rd edn. Springer, New York (2002).
    • (2002) Algebra
    • Lang, S.1
  • 40
    • 0001018621 scopus 로고
    • Number of points of varieties in finite fields
    • Lang, S., Weil, A.: Number of points of varieties in finite fields. Am. J. Math. 76, 819-827 (1954).
    • (1954) Am. J. Math. , vol.76 , pp. 819-827
    • Lang, S.1    Weil, A.2
  • 42
    • 0002093056 scopus 로고
    • Cayley graphs: Eigenvalues, expanders and random walks
    • London Math. Soc. Lecture Note Ser, P. Rowbinson (Ed.), Cambridge: Cambridge University Press
    • Lubotzky, A.: Cayley graphs: eigenvalues, expanders and random walks. In: Rowbinson, P. (ed.) Surveys in Combinatorics. London Math. Soc. Lecture Note Ser., vol. 218, pp. 155-189. Cambridge University Press, Cambridge (1995).
    • (1995) Surveys in Combinatorics , vol.218 , pp. 155-189
    • Lubotzky, A.1
  • 47
    • 0002256750 scopus 로고
    • Ein algebraisches Kriterium für absolute Irreduzibilität
    • Noether, E.: Ein algebraisches Kriterium für absolute Irreduzibilität. Math. Ann. 85, 26-40 (1922).
    • (1922) Math. Ann. , vol.85 , pp. 26-40
    • Noether, E.1
  • 49
    • 84941844016 scopus 로고
    • Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres
    • Sansuc, J. J.: Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres. J. Reine Angew. Math. 327, 12-80 (1981).
    • (1981) J. Reine Angew. Math. , vol.327 , pp. 12-80
    • Sansuc, J.J.1
  • 50
    • 23444458006 scopus 로고    scopus 로고
    • What is an expander?
    • Sarnak, P.: What is an expander? Not. Am. Math. Soc. 51, 762-763 (2004).
    • (2004) Not. Am. Math. Soc. , vol.51 , pp. 762-763
    • Sarnak, P.1
  • 51
    • 33746023130 scopus 로고    scopus 로고
    • Notes on the generalized Ramanujan conjectures
    • Sarnak, P.: Notes on the generalized Ramanujan conjectures. Clay Math. Proc. 4, 659-685 (2005).
    • (2005) Clay Math. Proc. , vol.4 , pp. 659-685
    • Sarnak, P.1
  • 54
    • 84974005758 scopus 로고
    • Bounds for multiplicities of automorphic representations
    • Sarnak, P., Xue, X.: Bounds for multiplicities of automorphic representations. Duke Math. J. 64, 207-227 (1991).
    • (1991) Duke Math. J. , vol.64 , pp. 207-227
    • Sarnak, P.1    Xue, X.2
  • 55
    • 0000579840 scopus 로고
    • Sur certaines hypotheses concernant les nombres premiers
    • Schinzel, A., Sierpinksi, W.: Sur certaines hypotheses concernant les nombres premiers. Acta Arith. 4, 185-208 (1958).
    • (1958) Acta Arith. , vol.4 , pp. 185-208
    • Schinzel, A.1    Sierpinksi, W.2
  • 57
    • 0002739011 scopus 로고
    • On the estimation of Fourier coefficients of modular forms
    • Providence: AMS
    • Selberg, A.: On the estimation of Fourier coefficients of modular forms. In: Proc. Symp. Pure Math., Vol. VII, pp. 1-15. AMS, Providence (1965).
    • (1965) Proc. Symp. Pure Math. , vol.VII , pp. 1-15
    • Selberg, A.1
  • 59
    • 67349224027 scopus 로고    scopus 로고
    • Product sets estimates for non-commutative groups
    • Tao, T.: Product sets estimates for non-commutative groups. Combinatorica 28, 547-594 (2008).
    • (2008) Combinatorica , vol.28 , pp. 547-594
    • Tao, T.1
  • 60
    • 33845302486 scopus 로고    scopus 로고
    • Cambridge: Cambridge University Press
    • Tao, T., Vu, V.: Additive Combinatorics. Cambridge University Press, Cambridge (2006).
    • (2006) Additive Combinatorics
    • Tao, T.1    Vu, V.2
  • 61
    • 0000644892 scopus 로고
    • Free subgroups in linear groups
    • Tits, J.: Free subgroups in linear groups. J. Algebra 20, 250-270 (1972).
    • (1972) J. Algebra , vol.20 , pp. 250-270
    • Tits, J.1
  • 63
    • 0000296996 scopus 로고
    • Representations of an odd number as a sum of three primes
    • Vinogradov, I. M.: Representations of an odd number as a sum of three primes. Dokl. Akad. Nauk SSSR 15, 291-294 (1937).
    • (1937) Dokl. Akad. Nauk SSSR , vol.15 , pp. 291-294
    • Vinogradov, I.M.1
  • 64
    • 0000447570 scopus 로고
    • Strong approximation for Zariski-dense subgroups of semisimple algebraic groups
    • Weisfeiler, B.: Strong approximation for Zariski-dense subgroups of semisimple algebraic groups. Ann. Math. 120(2), 271-231 (1984).
    • (1984) Ann. Math. , vol.120 , Issue.2 , pp. 271-231
    • Weisfeiler, B.1


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