An inverse theorem for the Gowers $U^{s+1}[N]$-norm

Abstract

We prove the inverse conjecture for the Gowers $U^{s+1}[N]$-norm for all $s \geq 1$; this is new for $s \geq 4$. More precisely, we establish that if $f : [N] \rightarrow [-1,1]$ is a function with $\Vert f \Vert_{U^{s+1}[N]} \geq \delta$, then there is a bounded-complexity $s$-step nilsequence $F(g(n)\Gamma)$ that correlates with $f$, where the bounds on the complexity and correlation depend only on $s$ and $\delta$. From previous results, this conjecture implies the Hardy-Littlewood prime tuples conjecture for any linear system of finite complexity.

  • [akklr] Go to document N. Alon, T. Kaufman, M. Krivelevich, S. Litsyn, and D. Ron, "Testing low-degree polynomials over ${ GF}(2)$," in Approximation, Randomization, and Combinatorial Optimization, New York: Springer-Verlag, 2003, vol. 2764, pp. 188-199.
    @incollection {akklr, MRKEY = {2080792},
      AUTHOR = {Alon, Noga and Kaufman, Tali and Krivelevich, Michael and Litsyn, Simon and Ron, Dana},
      TITLE = {Testing low-degree polynomials over {${\rm GF}(2)$}},
      BOOKTITLE = {Approximation, Randomization, and Combinatorial Optimization},
      SERIES = {Lecture Notes in Comput. Sci.},
      VOLUME = {2764},
      PAGES = {188--199},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2003},
      MRCLASS = {68W20 (68W40)},
      MRNUMBER = {2080792},
      DOI = {10.1007/978-3-540-45198-3_17},
      ZBLNUMBER = {05712664},
      }
  • [balog] Go to document A. Balog and E. Szemerédi, "A statistical theorem of set addition," Combinatorica, vol. 14, iss. 3, pp. 263-268, 1994.
    @article {balog, MRKEY = {1305895},
      AUTHOR = {Balog, Antal and Szemer{é}di, Endre},
      TITLE = {A statistical theorem of set addition},
      JOURNAL = {Combinatorica},
      FJOURNAL = {Combinatorica. An International Journal on Combinatorics and the Theory of Computing},
      VOLUME = {14},
      YEAR = {1994},
      NUMBER = {3},
      PAGES = {263--268},
      ISSN = {0209-9683},
      CODEN = {COMBDI},
      MRCLASS = {11B25 (11B75)},
      MRNUMBER = {1305895},
      MRREVIEWER = {Anne Ludington Young},
      DOI = {10.1007/BF01212974},
      ZBLNUMBER = {0812.11017},
      }
  • [bl] Go to document V. Bergelson and A. Leibman, "Distribution of values of bounded generalized polynomials," Acta Math., vol. 198, iss. 2, pp. 155-230, 2007.
    @article {bl, MRKEY = {2318563},
      AUTHOR = {Bergelson, Vitaly and Leibman, Alexander},
      TITLE = {Distribution of values of bounded generalized polynomials},
      JOURNAL = {Acta Math.},
      FJOURNAL = {Acta Mathematica},
      VOLUME = {198},
      YEAR = {2007},
      NUMBER = {2},
      PAGES = {155--230},
      ISSN = {0001-5962},
      CODEN = {ACMAA8},
      MRCLASS = {11K31 (11J54)},
      MRNUMBER = {2318563},
      MRREVIEWER = {Alexander Gorodnik},
      DOI = {10.1007/s11511-007-0015-y},
      ZBLNUMBER = {1137.37005},
      }
  • [bhk] Go to document V. Bergelson, B. Host, and B. Kra, "Multiple recurrence and nilsequences," Invent. Math., vol. 160, iss. 2, pp. 261-303, 2005.
    @article {bhk, MRKEY = {2138068},
      AUTHOR = {Bergelson, Vitaly and Host, Bernard and Kra, Bryna},
      TITLE = {Multiple recurrence and nilsequences},
      NOTE = {with an appendix by Imre Ruzsa},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {160},
      YEAR = {2005},
      NUMBER = {2},
      PAGES = {261--303},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {37A30 (05D10 28D05 37A05)},
      MRNUMBER = {2138068},
      MRREVIEWER = {Randall McCutcheon},
      DOI = {10.1007/s00222-004-0428-6},
      ZBLNUMBER = {1087.28007},
      }
  • [bergelson-tao-ziegler] Go to document V. Bergelson, T. Tao, and T. Ziegler, "An inverse theorem for the uniformity seminorms associated with the action of $\Bbb F^\infty_p$," Geom. Funct. Anal., vol. 19, iss. 6, pp. 1539-1596, 2010.
    @article {bergelson-tao-ziegler, MRKEY = {2594614},
      AUTHOR = {Bergelson, Vitaly and Tao, Terence and Ziegler, Tamar},
      TITLE = {An inverse theorem for the uniformity seminorms associated with the action of {$\Bbb F\sp \infty\sb p$}},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {19},
      YEAR = {2010},
      NUMBER = {6},
      PAGES = {1539--1596},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {37A35 (28D05)},
      MRNUMBER = {2594614},
      MRREVIEWER = {Randall McCutcheon},
      DOI = {10.1007/s00039-010-0051-1},
      ZBLNUMBER = {1189.37007},
      }
  • [bogolyubov] N. N. Bogolyubov, "Sur quelques propriétés arithmétiques des presque-périodes," Ann. Chaire Math. Phys. Kiev, vol. 4, pp. 185-194, 1939.
    @article{bogolyubov,
      author={Bogolyubov, N. N.},
      TITLE={Sur quelques propriétés arithmétiques des presque-périodes},
      JOURNAL={Ann. Chaire Math. Phys. Kiev},
      VOLUME={4},
      YEAR={1939},
      PAGES={185--194},
     }
  • [bourbaki] N. Bourbaki, Groupes et algèbres de Lie, vol. I.
    @misc{bourbaki,
      author={Bourbaki, N.},
      TITLE={Groupes et algèbres de {L}ie, vol. {I}},
      NOTE={preprint},
     }
  • [bourgain-sumsets] J. Bourgain, "On arithmetic progressions in sums of sets of integers," in A Tribute to Paul Erdős, Cambridge: Cambridge Univ. Press, 1990, pp. 105-109.
    @incollection {bourgain-sumsets, MRKEY = {1117007},
      AUTHOR = {Bourgain, J.},
      TITLE = {On arithmetic progressions in sums of sets of integers},
      BOOKTITLE = {A Tribute to {P}aul {E}rdős},
      PAGES = {105--109},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1990},
      MRCLASS = {11B25 (11N64)},
      MRNUMBER = {1117007},
      MRREVIEWER = {Armel Mercier},
      ZBLNUMBER = {0715.11006},
      }
  • [bourgain-3aps] Go to document J. Bourgain, "On triples in arithmetic progression," Geom. Funct. Anal., vol. 9, iss. 5, pp. 968-984, 1999.
    @article {bourgain-3aps, MRKEY = {1726234},
      AUTHOR = {Bourgain, J.},
      TITLE = {On triples in arithmetic progression},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {9},
      YEAR = {1999},
      NUMBER = {5},
      PAGES = {968--984},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {11P55 (11B25 11N64)},
      MRNUMBER = {1726234},
      MRREVIEWER = {Serge{\u\i} V. Konyagin},
      DOI = {10.1007/s000390050105},
      ZBLNUMBER = {0959.11004},
      }
  • [conze] J. Conze and E. Lesigne, "Sur un théorème ergodique pour des mesures diagonales," C. R. Acad. Sci. Paris Sér. I Math., vol. 306, iss. 12, pp. 491-493, 1988.
    @article {conze, MRKEY = {0939438},
      AUTHOR = {Conze, Jean-Pierre and Lesigne, Emmanuel},
      TITLE = {Sur un théorème ergodique pour des mesures diagonales},
      JOURNAL = {C. R. Acad. Sci. Paris Sér. I Math.},
      FJOURNAL = {Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique},
      VOLUME = {306},
      YEAR = {1988},
      NUMBER = {12},
      PAGES = {491--493},
      ISSN = {0249-6291},
      CODEN = {CASMEI},
      MRCLASS = {22D40 (28D05)},
      MRNUMBER = {0939438},
      MRREVIEWER = {Pierre Michel},
      ZBLNUMBER = {0641.28010},
      }
  • [fhk] Go to document N. Frantzikinakis, B. Host, and B. Kra, "Multiple recurrence and convergence for sequences related to the prime numbers," J. Reine Angew. Math., vol. 611, pp. 131-144, 2007.
    @article {fhk, MRKEY = {2361086},
      AUTHOR = {Frantzikinakis, Nikos and Host, Bernard and Kra, Bryna},
      TITLE = {Multiple recurrence and convergence for sequences related to the prime numbers},
      JOURNAL = {J. Reine Angew. Math.},
      FJOURNAL = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      VOLUME = {611},
      YEAR = {2007},
      PAGES = {131--144},
      ISSN = {0075-4102},
      CODEN = {JRMAA8},
      MRCLASS = {11B25 (11A41 28D05 37A05)},
      MRNUMBER = {2361086},
      MRREVIEWER = {Samuel S. Wagstaff, Jr.},
      DOI = {10.1515/CRELLE.2007.076},
      ZBLNUMBER = {1126.37005},
      }
  • [furstenberg] Go to document H. Furstenberg, "Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions," J. Analyse Math., vol. 31, pp. 204-256, 1977.
    @article {furstenberg, MRKEY = {0498471},
      AUTHOR = {Furstenberg, Harry},
      TITLE = {Ergodic behavior of diagonal measures and a theorem of {S}zemerédi on arithmetic progressions},
      JOURNAL = {J. Analyse Math.},
      FJOURNAL = {Journal d'Analyse Mathématique},
      VOLUME = {31},
      YEAR = {1977},
      PAGES = {204--256},
      ISSN = {0021-7670},
      MRCLASS = {10L10 (10K10 28A65)},
      MRNUMBER = {0498471},
      MRREVIEWER = {Francois Aribaud},
      ZBLNUMBER = {0347.28016},
      DOI = {10.1007/BF02813304},
     }
  • [furst] H. Furstenberg, "Nonconventional ergodic averages," in The Legacy of John von Neumann, Providence, RI: Amer. Math. Soc., 1990, vol. 50, pp. 43-56.
    @incollection {furst, MRKEY = {1067751},
      AUTHOR = {Furstenberg, Hillel},
      TITLE = {Nonconventional ergodic averages},
      BOOKTITLE = {The Legacy of {J}ohn von {N}eumann},
      VENUE={{H}empstead, {NY},
      1988},
      SERIES = {Proc. Sympos. Pure Math.},
      VOLUME = {50},
      PAGES = {43--56},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1990},
      MRCLASS = {28D05 (47A35)},
      MRNUMBER = {1067751},
      MRREVIEWER = {M. A. Akcoglu},
      ZBLNUMBER = {0711.28006},
      }
  • [fw-char] Go to document H. Furstenberg and B. Weiss, "A mean ergodic theorem for $(1/N)\sum^N_{n=1}f(T^nx)g(T^{n^2}x)$," in Convergence in Ergodic Theory and Probability, Berlin: de Gruyter, 1996, vol. 5, pp. 193-227.
    @incollection {fw-char, MRKEY = {1412607},
      AUTHOR = {Furstenberg, Hillel and Weiss, Benjamin},
      TITLE = {A mean ergodic theorem for {$(1/N)\sum\sp N\sb {n=1}f(T\sp nx)g(T\sp {n\sp 2}x)$}},
      BOOKTITLE = {Convergence in Ergodic Theory and Probability},
      VENUE={{C}olumbus, {OH},
      1993},
      SERIES = {Ohio State Univ. Math. Res. Inst. Publ.},
      VOLUME = {5},
      PAGES = {193--227},
      PUBLISHER = {de Gruyter},
      ADDRESS = {Berlin},
      YEAR = {1996},
      MRCLASS = {28D05},
      MRNUMBER = {1412607},
      MRREVIEWER = {Idris Assani},
      ZBLNUMBER = {0869.28010},
      DOI = {10.1515/9783110889383.193},
     }
  • [godel] Go to document K. Gödel, "The consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory," Proc. Nat. Acad. Sci., vol. 24, pp. 556-557, 1938.
    @article{godel,
      author={Gödel, K.},
      TITLE={The consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory},
      JOURNAL={Proc. Nat. Acad. Sci.},
      VOLUME={24},
      YEAR={1938},
      PAGES={556--557},
      ZBLNUMBER = {0020.29701},
      URL = {http://www.pnas.org/content/24/12/556.full.pdf+html},
     }
  • [gowers-4aps] Go to document W. T. Gowers, "A new proof of Szemerédi’s theorem for arithmetic progressions of length four," Geom. Funct. Anal., vol. 8, iss. 3, pp. 529-551, 1998.
    @article {gowers-4aps, MRKEY = {1631259},
      AUTHOR = {Gowers, W. T.},
      TITLE = {A new proof of {S}zemerédi's theorem for arithmetic progressions of length four},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {8},
      YEAR = {1998},
      NUMBER = {3},
      PAGES = {529--551},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {11B25 (11N13)},
      MRNUMBER = {1631259},
      MRREVIEWER = {D. R. Heath-Brown},
      DOI = {10.1007/s000390050065},
      ZBLNUMBER = {0907.11005},
      }
  • [gowers-longaps] Go to document W. T. Gowers, "A new proof of Szemerédi’s theorem," Geom. Funct. Anal., vol. 11, iss. 3, pp. 465-588, 2001.
    @article {gowers-longaps, MRKEY = {1844079},
      AUTHOR = {Gowers, W. T.},
      TITLE = {A new proof of {S}zemerédi's theorem},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {11},
      YEAR = {2001},
      NUMBER = {3},
      PAGES = {465--588},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {11B25 (11K38 11K45)},
      MRNUMBER = {1844079},
      MRREVIEWER = {Hillel Furstenberg},
      DOI = {10.1007/s00039-001-0332-9},
      ZBLNUMBER = {1028.11005},
      }
  • [green-sumset] Go to document B. Green, "Arithmetic progressions in sumsets," Geom. Funct. Anal., vol. 12, iss. 3, pp. 584-597, 2002.
    @article {green-sumset, MRKEY = {1924373},
      AUTHOR = {Green, Benjamin},
      TITLE = {Arithmetic progressions in sumsets},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {12},
      YEAR = {2002},
      NUMBER = {3},
      PAGES = {584--597},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {11P70 (11B05 11K99)},
      MRNUMBER = {1924373},
      MRREVIEWER = {Alain Plagne},
      DOI = {10.1007/s00039-002-8258-4},
      ZBLNUMBER = {1020.11009},
      }
  • [green-icm] Go to document B. Green, "Generalising the Hardy-Littlewood method for primes," in International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 373-399.
    @incollection {green-icm, MRKEY = {2275602},
      AUTHOR = {Green, Benjamin},
      TITLE = {Generalising the {H}ardy-{L}ittlewood method for primes},
      BOOKTITLE = {International {C}ongress of {M}athematicians. {V}ol. {II}},
      PAGES = {373--399},
      PUBLISHER = {Eur. Math. Soc., Zürich},
      YEAR = {2006},
      MRCLASS = {11B25 (11P55)},
      MRNUMBER = {2275602},
      MRREVIEWER = {John B. Friedlander},
      ZBLNUMBER = {1157.11007},
      DOI = {10.4171/022-2/17},
     }
  • [green-tao-longaps] Go to document B. Green and T. Tao, "The primes contain arbitrarily long arithmetic progressions," Ann. of Math., vol. 167, iss. 2, pp. 481-547, 2008.
    @article {green-tao-longaps, MRKEY = {2415379},
      AUTHOR = {Green, Benjamin and Tao, Terence},
      TITLE = {The primes contain arbitrarily long arithmetic progressions},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {167},
      YEAR = {2008},
      NUMBER = {2},
      PAGES = {481--547},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11N13 (11A41 11B25 37A45)},
      MRNUMBER = {2415379},
      MRREVIEWER = {Tamar Ziegler},
      DOI = {10.4007/annals.2008.167.481},
      ZBLNUMBER = {1191.11025},
      }
  • [green-tao-u3inverse] B. Green and T. Tao, "An inverse theorem for the Gowers ${U}^3$-norm, with applications," Proc. Edinburgh Math. Soc., vol. 51, pp. 71-153, 2008.
    @article{green-tao-u3inverse,
      author = {Green, Benjamin and Tao, Terence},
      TITLE = {An inverse theorem for the {G}owers ${U}^3$-norm, with applications},
      JOURNAL={Proc. Edinburgh Math. Soc.},
      VOLUME={51},
      YEAR = {2008},
      MRNUMBER = {2391635},
      ZBLNUMBER = {1202.11013},
      PAGES={71--153},
     }
  • [green-tao-u3mobius] Go to document B. Green and T. Tao, "Quadratic uniformity of the Möbius function," Ann. Inst. Fourier $($Grenoble$)$, vol. 58, iss. 6, pp. 1863-1935, 2008.
    @article {green-tao-u3mobius, MRKEY = {2473624},
      AUTHOR = {Green, Benjamin and Tao, Terence},
      TITLE = {Quadratic uniformity of the {M}öbius function},
      JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
      FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
      VOLUME = {58},
      YEAR = {2008},
      NUMBER = {6},
      PAGES = {1863--1935},
      ISSN = {0373-0956},
      CODEN = {AIFUA7},
      MRCLASS = {11B30 (11A25)},
      MRNUMBER = {2473624},
      MRREVIEWER = {Tamar Ziegler},
      URL = {http://aif.cedram.org/item?id=AIF_2008__58_6_1863_0},
      }
  • [green-tao-linearprimes] Go to document B. Green and T. Tao, "Linear equations in primes," Ann. of Math., vol. 171, iss. 3, pp. 1753-1850, 2010.
    @article {green-tao-linearprimes, MRKEY = {2680398},
      AUTHOR = {Green, Benjamin and Tao, Terence},
      TITLE = {Linear equations in primes},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {171},
      YEAR = {2010},
      NUMBER = {3},
      PAGES = {1753--1850},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11N13 (11B30 11P32)},
      MRNUMBER = {2680398},
      MRREVIEWER = {Tamar Ziegler},
      DOI = {10.4007/annals.2010.171.1753},
      ZBLNUMBER = {05712763},
      }
  • [green-tao-nilratner] Go to document B. Green and T. Tao, "The quantitative behaviour of polynomial orbits on nilmanifolds," Ann. of Math., vol. 175, iss. 2, pp. 465-540, 2012.
    @article{green-tao-nilratner, MRKEY={2877065},
      AUTHOR = {Green, Benjamin and Tao, Terence},
      TITLE = {The quantitative behaviour of polynomial orbits on nilmanifolds},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {175},
      SORTYEAR={2011},
      YEAR = {2012},
      NUMBER = {2},
      PAGES = {465--540},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {37Axx (22Exx)},
      MRNUMBER = {2877065},
      ZBLNUMBER = {06024997},
      DOI = {10.4007/annals.2012.175.2.2},
      }
  • [green-tao-finite] B. Green and T. Tao, "The distribution of polynomials over finite fields, with applications to the Gowers norms," Contrib. Discrete Math., vol. 4, iss. 2, pp. 1-36, 2009.
    @article {green-tao-finite, MRKEY = {2592422},
      AUTHOR = {Green, Benjamin and Tao, Terence},
      TITLE = {The distribution of polynomials over finite fields, with applications to the {G}owers norms},
      JOURNAL = {Contrib. Discrete Math.},
      FJOURNAL = {Contributions to Discrete Mathematics},
      VOLUME = {4},
      YEAR = {2009},
      NUMBER = {2},
      PAGES = {1--36},
      ISSN = {1715-0868},
      MRCLASS = {11B30 (11B75 11K36)},
      MRNUMBER = {2592422},
      MRREVIEWER = {Serge{\u\i} V. Konyagin},
      ZBLNUMBER = {1225.11017},
      }
  • [green-tao-mobiusnilsequences] Go to document B. Green and T. Tao, "The Möbius function is strongly orthogonal to nilsequences," Ann. of Math., vol. 175, iss. 2, pp. 541-566, 2012.
    @article{green-tao-mobiusnilsequences, MRKEY={2877066},
      AUTHOR = {Green, Benjamin and Tao, Terence},
      TITLE = {The {M}öbius function is strongly orthogonal to nilsequences},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {175},
      YEAR = {2012},
      NUMBER = {2},
      PAGES = {541--566},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {37Axx (11A25 22Exx)},
      MRNUMBER = {2877066},
      ZBLNUMBER = {06024998},
      DOI = {10.4007/annals.2012.175.2.3},
      }
  • [green-tao-arithmetic-regularity] Go to document B. Green and T. Tao, "An arithmetic regularity lemma, an associated counting lemma, and applications," in An Irregular Mind, Budapest: János Bolyai Math. Soc., 2010, vol. 21, pp. 261-334.
    @incollection {green-tao-arithmetic-regularity, MRKEY = {2815606},
      AUTHOR = {Green, Benjamin and Tao, Terence},
      TITLE = {An arithmetic regularity lemma, an associated counting lemma, and applications},
      BOOKTITLE = {An Irregular Mind},
      SERIES = {Bolyai Soc. Math. Stud.},
      VOLUME = {21},
      PAGES = {261--334},
      PUBLISHER = {János Bolyai Math. Soc.},
      ADDRESS = {Budapest},
      YEAR = {2010},
      MRCLASS = {11B30 (05D05)},
      MRNUMBER = {2815606},
      DOI = {10.1007/978-3-642-14444-8_7},
      ZBLNUMBER = {1222.11015},
      }
  • [u4-inverse] Go to document B. Green, T. Tao, and T. Ziegler, "An inverse theorem for the Gowers $U^4$-norm," Glasg. Math. J., vol. 53, iss. 1, pp. 1-50, 2011.
    @article {u4-inverse, MRKEY = {2747135},
      AUTHOR = {Green, Benjamin and Tao, Terence and Ziegler, Tamar},
      TITLE = {An inverse theorem for the {G}owers {$U\sp 4$}-norm},
      JOURNAL = {Glasg. Math. J.},
      FJOURNAL = {Glasgow Mathematical Journal},
      VOLUME = {53},
      YEAR = {2011},
      NUMBER = {1},
      PAGES = {1--50},
      ISSN = {0017-0895},
      MRCLASS = {11B30 (11N13 11P70)},
      MRNUMBER = {2747135},
      MRREVIEWER = {Julia Wolf},
      DOI = {10.1017/S0017089510000546},
      ZBLNUMBER = {05843247},
      }
  • [gtz-announce] Go to document B. Green, T. Tao, and T. Ziegler, "An inverse theorem for the Gowers ${U}^{s+1}[{N}]$-norm," Electron. Res. Announc. Math. Sci., vol. 18, pp. 69-90, 2011.
    @article{gtz-announce,
      author = {Green, Benjamin and Tao, Terence and Ziegler, Tamar},
      TITLE = {An inverse theorem for the {G}owers ${U}^{s+1}[{N}]$-norm},
      JOURNAL = {Electron. Res. Announc. Math. Sci.},
      FJOURNAL = {Electronic Research Announcements in Mathematical Sciences},
      VOLUME = {18},
      YEAR = {2011},
      PAGES = {69--90},
      ISSN = {1935-9179},
      MRCLASS = {11B30},
      MRNUMBER = {2817840},
      MRREVIEWER = {Andreas O. Bender},
      ZBLNUMBER = {05920419},
      DOI = {10.3934/era.2011.18.69},
      URL = {http://dx.doi.org/10.3934/era.2011.18.69},
      }%%%%
  • [ha1] Go to document I. J. Håland, Uniform distribution of generalized polynomials, 1992.
    @misc{ha1,
      author = {H{\aa}land, Inger Johanne},
      TITLE = {Uniform distribution of generalized polynomials},
      NOTE={Ph.D. thesis, The Ohio State University},
      YEAR={1992},
      ZBLNUMBER = {0797.11064},
      DOI = {10.1017/S0013091505000325},
     }
  • [ha2] Go to document I. J. Håland, "Uniform distribution of generalized polynomials," J. Number Theory, vol. 45, iss. 3, pp. 327-366, 1993.
    @article {ha2, MRKEY = {1247389},
      AUTHOR = {H{\aa}land, Inger Johanne},
      TITLE = {Uniform distribution of generalized polynomials},
      JOURNAL = {J. Number Theory},
      FJOURNAL = {Journal of Number Theory},
      VOLUME = {45},
      YEAR = {1993},
      NUMBER = {3},
      PAGES = {327--366},
      ISSN = {0022-314X},
      CODEN = {JNUTA9},
      MRCLASS = {11K06 (11J71)},
      MRNUMBER = {1247389},
      DOI = {10.1006/jnth.1993.1082},
      ZBLNUMBER = {0797.11064},
     }
  • [ha3] Go to document I. J. Håland, "Uniform distribution of generalized polynomials of the product type," Acta Arith., vol. 67, iss. 1, pp. 13-27, 1994.
    @article {ha3, MRKEY = {1292518},
      AUTHOR = {H{\aa}land, Inger Johanne},
      TITLE = {Uniform distribution of generalized polynomials of the product type},
      JOURNAL = {Acta Arith.},
      FJOURNAL = {Acta Arithmetica},
      VOLUME = {67},
      YEAR = {1994},
      NUMBER = {1},
      PAGES = {13--27},
      ISSN = {0065-1036},
      CODEN = {AARIA9},
      MRCLASS = {11K06},
      MRNUMBER = {1292518},
      MRREVIEWER = {Robert F. Tichy},
      ZBLNUMBER = {0805.11054},
      URL ={http://pldml.icm.edu.pl/mathbwn/element/bwmeta1.element.bwnjournal-article-aav67i1p13bwm?q=18dbe055-5edc-4f3a-b226-91440b2787ce$1&qt=IN_PAGE,
     
  • [hak] I. J. Håland and D. E. Knuth, "Polynomials involving the floor function," Math. Scand., vol. 76, iss. 2, pp. 194-200, 1995.
    @article {hak, MRKEY = {1354576},
      AUTHOR = {H{\aa}land, Inger Johanne and Knuth, Donald E.},
      TITLE = {Polynomials involving the floor function},
      JOURNAL = {Math. Scand.},
      FJOURNAL = {Mathematica Scandinavica},
      VOLUME = {76},
      YEAR = {1995},
      NUMBER = {2},
      PAGES = {194--200},
      ISSN = {0025-5521},
      CODEN = {MTSCAN},
      MRCLASS = {11K06},
      MRNUMBER = {1354576},
      MRREVIEWER = {Robert F. Tichy},
      }
  • [hk0] Go to document B. Host and B. Kra, "Convergence of Conze-Lesigne averages," Ergodic Theory Dynam. Systems, vol. 21, iss. 2, pp. 493-509, 2001.
    @article {hk0, MRKEY = {1827115},
      AUTHOR = {Host, Bernard and Kra, Bryna},
      TITLE = {Convergence of {C}onze-{L}esigne averages},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {21},
      YEAR = {2001},
      NUMBER = {2},
      PAGES = {493--509},
      ISSN = {0143-3857},
      MRCLASS = {28D05 (37A30)},
      MRNUMBER = {1827115},
      MRREVIEWER = {James H. Olsen},
      DOI = {10.1017/S0143385701001249},
      ZBLNUMBER = {0995.37003},
      }
  • [hk1] B. Host and B. Kra, "Averaging along cubes," in Modern Dynamical Systems and Applications, Cambridge: Cambridge Univ. Press, 2004, pp. 123-144.
    @incollection {hk1, MRKEY = {2090768},
      AUTHOR = {Host, Bernard and Kra, Bryna},
      TITLE = {Averaging along cubes},
      BOOKTITLE = {Modern Dynamical Systems and Applications},
      PAGES = {123--144},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2004},
      MRCLASS = {37A05 (28D05 37A15)},
      MRNUMBER = {2090768},
      MRREVIEWER = {James H. Olsen},
      ZBLNUMBER = {1147.37301},
      }
  • [host-kra] Go to document B. Host and B. Kra, "Nonconventional ergodic averages and nilmanifolds," Ann. of Math., vol. 161, iss. 1, pp. 397-488, 2005.
    @article {host-kra, MRKEY = {2150389},
      AUTHOR = {Host, Bernard and Kra, Bryna},
      TITLE = {Nonconventional ergodic averages and nilmanifolds},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {161},
      YEAR = {2005},
      NUMBER = {1},
      PAGES = {397--488},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {37A05 (28D05)},
      MRNUMBER = {2150389},
      MRREVIEWER = {Randall McCutcheon},
      DOI = {10.4007/annals.2005.161.397},
      ZBLNUMBER = {1077.37002},
      }
  • [host-kra-uniformity] Go to document B. Host and B. Kra, "Uniformity seminorms on $\ell^\infty$ and applications," J. Anal. Math., vol. 108, pp. 219-276, 2009.
    @article {host-kra-uniformity, MRKEY = {2544760},
      AUTHOR = {Host, Bernard and Kra, Bryna},
      TITLE = {Uniformity seminorms on {$\ell\sp \infty$} and applications},
      JOURNAL = {J. Anal. Math.},
      FJOURNAL = {Journal d'Analyse Mathématique},
      VOLUME = {108},
      YEAR = {2009},
      PAGES = {219--276},
      ISSN = {0021-7670},
      CODEN = {JOAMAV},
      MRCLASS = {11B30 (11B25 28D05 37A45)},
      MRNUMBER = {2544760},
      MRREVIEWER = {Serge{\u\i} V. Konyagin},
      DOI = {10.1007/s11854-009-0024-1},
      ZBLNUMBER = {1183.37011},
     }
  • [host-kra-nilseq] Go to document B. Host, B. Kra, and A. Maass, "Nilsequences and a structure theorem for topological dynamical systems," Adv. Math., vol. 224, iss. 1, pp. 103-129, 2010.
    @article {host-kra-nilseq, MRKEY = {2600993},
      AUTHOR = {Host, Bernard and Kra, Bryna and Maass, Alejandro},
      TITLE = {Nilsequences and a structure theorem for topological dynamical systems},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {224},
      YEAR = {2010},
      NUMBER = {1},
      PAGES = {103--129},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {37B05 (37A05 54H20)},
      MRNUMBER = {2600993},
      MRREVIEWER = {Artur Siemaszko},
      DOI = {10.1016/j.aim.2009.11.009},
      ZBLNUMBER = {1203.37022},
      }
  • [hrush] Go to document E. Hrushovski, "Totally categorical structures," Trans. Amer. Math. Soc., vol. 313, iss. 1, pp. 131-159, 1989.
    @article {hrush, MRKEY = {0943605},
      AUTHOR = {Hrushovski, Ehud},
      TITLE = {Totally categorical structures},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {313},
      YEAR = {1989},
      NUMBER = {1},
      PAGES = {131--159},
      ISSN = {0002-9947},
      CODEN = {TAMTAM},
      MRCLASS = {03C45 (03C35)},
      MRNUMBER = {0943605},
      MRREVIEWER = {U. Felgner},
      DOI = {10.2307/2001069},
      ZBLNUMBER = {0674.03009},
      }
  • [jin] R. Jin, Freiman’s Conjecture and Nonstandard Methods.
    @misc{jin,
      author={Jin, R.},
      TITLE={Freiman's Conjecture and Nonstandard Methods},
      NOTE={preprint},
     }
  • [kauf] T. Kaufman and S. Lovett, Worst Case to Average Case Reductions for Polynomials.
    @misc{kauf,
      author={Kaufman, T. and Lovett, S.},
      TITLE={Worst Case to Average Case Reductions for Polynomials},
      NOTE={preprint},
     }
  • [lazard] Go to document M. Lazard, "Sur les groupes nilpotents et les anneaux de Lie," Ann. Sci. Ecole Norm. Sup., vol. 71, pp. 101-190, 1954.
    @article {lazard, MRKEY = {0088496},
      AUTHOR = {Lazard, Michel},
      TITLE = {Sur les groupes nilpotents et les anneaux de {L}ie},
      JOURNAL = {Ann. Sci. Ecole Norm. Sup.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Troisième Série},
      VOLUME = {71},
      YEAR = {1954},
      PAGES = {101--190},
      ISSN = {0012-9593},
      MRCLASS = {20.0X},
      MRNUMBER = {0088496},
      MRREVIEWER = {B. H. Neumann},
      ZBLNUMBER = {0055.25103},
      URL = {http://www.numdam.org/item?id=ASENS_1954_3_71_2_101_0},
     }
  • [leibman-group-1] Go to document A. Leibman, "Polynomial sequences in groups," J. Algebra, vol. 201, iss. 1, pp. 189-206, 1998.
    @article {leibman-group-1, MRKEY = {1608723},
      AUTHOR = {Leibman, A.},
      TITLE = {Polynomial sequences in groups},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {201},
      YEAR = {1998},
      NUMBER = {1},
      PAGES = {189--206},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {20F14},
      MRNUMBER = {1608723},
      MRREVIEWER = {Mohammad Reza R. Moghaddam},
      DOI = {10.1006/jabr.1997.7269},
      ZBLNUMBER = {0908.20029},
      }
  • [leibman-group-2] Go to document A. Leibman, "Polynomial mappings of groups," Israel J. Math., vol. 129, pp. 29-60, 2002.
    @article {leibman-group-2, MRKEY = {1910931},
      AUTHOR = {Leibman, A.},
      TITLE = {Polynomial mappings of groups},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {129},
      YEAR = {2002},
      PAGES = {29--60},
      ISSN = {0021-2172},
      CODEN = {ISJMAP},
      MRCLASS = {20F18 (20K30 37A25 43A07 47D03)},
      MRNUMBER = {1910931},
      MRREVIEWER = {Tullio G. Ceccherini-Silberstein},
      DOI = {10.1007/BF02773152},
      ZBLNUMBER = {1007.20035},
      }
  • [leibman-poly] Go to document A. Leibman, "Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold," Ergodic Theory Dynam. Systems, vol. 25, iss. 1, pp. 201-213, 2005.
    @article {leibman-poly, MRKEY = {2122919},
      AUTHOR = {Leibman, A.},
      TITLE = {Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold},
      JOURNAL = {Ergodic Theory Dynam. Systems},
      FJOURNAL = {Ergodic Theory and Dynamical Systems},
      VOLUME = {25},
      YEAR = {2005},
      NUMBER = {1},
      PAGES = {201--213},
      ISSN = {0143-3857},
      MRCLASS = {37A17 (22F30 28D15)},
      MRNUMBER = {2122919},
      MRREVIEWER = {Alexander Gorodnik},
      DOI = {10.1017/S0143385704000215},
      ZBLNUMBER = {1080.37003},
      }
  • [leibman] Go to document A. Leibman, "A canonical form and the distribution of values of generalised polynomials," Israel J. Math., vol. 188, pp. 131-176, 2012.
    @article{leibman,
      author = {Leibman, A.},
      TITLE = {A canonical form and the distribution of values of generalised polynomials},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {188},
      YEAR = {2012},
      PAGES = {131--176},
      DOI = {10.1007/s11856-011-0097-2},
     }
  • [lesigne-nil] Go to document E. Lesigne, "Équations fonctionnelles, couplages de produits gauches et théorèmes ergodiques pour mesures diagonales," Bull. Soc. Math. France, vol. 121, iss. 3, pp. 315-351, 1993.
    @article {lesigne-nil, MRKEY = {1242635},
      AUTHOR = {Lesigne, E.},
      TITLE = {\'{E}quations fonctionnelles, couplages de produits gauches et théorèmes ergodiques pour mesures diagonales},
      JOURNAL = {Bull. Soc. Math. France},
      FJOURNAL = {Bulletin de la Société Mathématique de France},
      VOLUME = {121},
      YEAR = {1993},
      NUMBER = {3},
      PAGES = {315--351},
      ISSN = {0037-9484},
      CODEN = {BSMFAA},
      MRCLASS = {28D05},
      MRNUMBER = {1242635},
      MRREVIEWER = {Mariusz Lema{ń}czyk},
      URL = {http://smf4.emath.fr/Publications/Bulletin/121/html/smf_bull_121_315-351.html},
      ZBLNUMBER = {0798.28008},
      }
  • [malcev] Go to document A. I. Malcprimecev, "On a class of homogeneous spaces," Izvestiya Akad. Nauk. SSSR Ser. Mat., vol. 13, pp. 9-32, 1949.
    @article {malcev, MRKEY = {0028842},
      AUTHOR = {Mal{\cprime}cev, A. I.},
      TITLE = {On a class of homogeneous spaces},
      JOURNAL = {Izvestiya Akad. Nauk. SSSR Ser. Mat.},
      FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
      VOLUME = {13},
      YEAR = {1949},
      PAGES = {9--32},
      ISSN = {0373-2436},
      MRCLASS = {20.0X},
      MRNUMBER = {0028842},
      MRREVIEWER = {I. Kaplansky},
      ZBLNUMBER = {0034.01701},
      URL = {http://mi.mathnet.ru/eng/izv/v13/i1/p9},
     }
  • [mks] W. Magnus, A. Karrass, and D. Solitar, Combinatorial Group Theory. Presentations of Groups in Terms of Generators and Relations, second ed., Mineola, NY: Dover Publications, 2004.
    @book {mks, MRKEY = {2109550},
      AUTHOR = {Magnus, Wilhelm and Karrass, Abraham and Solitar, Donald},
      TITLE = {Combinatorial Group Theory. Presentations of Groups in Terms of Generators and Relations},
      EDITION = {second},
      PUBLISHER = {Dover Publications},
      ADDRESS = {Mineola, NY},
      YEAR = {2004},
      PAGES = {xii+444},
      ISBN = {0-486-43830-9},
      MRCLASS = {20E05 (05A15 20E06 20F12)},
      MRNUMBER = {2109550},
      ZBLNUMBER = {1130.20307},
      }
  • [ribet] Go to document K. A. Ribet, "On $\ell$-adic representations attached to modular forms," Invent. Math., vol. 28, pp. 245-275, 1975.
    @article {ribet, MRKEY = {0419358},
      AUTHOR = {Ribet, Kenneth A.},
      TITLE = {On {$\ell$}-adic representations attached to modular forms},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {28},
      YEAR = {1975},
      PAGES = {245--275},
      ISSN = {0020-9910},
      MRCLASS = {10D05 (12A55 14G25)},
      MRNUMBER = {0419358},
      MRREVIEWER = {F. Oort},
      ZBLNUMBER = {0302.10027},
      DOI = {10.1007/BF01425561},
     }
  • [sam] Go to document A. Samorodnitsky, "Low-degree tests at large distances," in STOC’07—Proceedings of the 39th Annual ACM Symposium on Theory of Computing, New York: ACM, 2007, pp. 506-515.
    @incollection {sam, MRKEY = {2402476},
      AUTHOR = {Samorodnitsky, Alex},
      TITLE = {Low-degree tests at large distances},
      BOOKTITLE = {S{TOC}'07---{P}roceedings of the 39th {A}nnual {ACM} {S}ymposium on {T}heory of {C}omputing},
      PAGES = {506--515},
      PUBLISHER = {ACM},
      ADDRESS = {New York},
      YEAR = {2007},
      MRCLASS = {68Q25 (68W20 94C10)},
      MRNUMBER = {2402476},
      DOI = {10.1145/1250790.1250864},
      ZBLNUMBER = {1232.68175},
      }
  • [szeg-1] B. Szegedy, Higher order Fourier analysis as an algebraic theory I.
    @misc{szeg-1,
      author={Szegedy, B.},
      TITLE={Higher order {F}ourier analysis as an algebraic theory {I}},
      ARXIV = {0903.0897},
     }
  • [szeg-2] B. Szegedy, Higher order Fourier analysis as an algebraic theory II.
    @misc{szeg-2,
      author={Szegedy, B.},
      TITLE={Higher order {F}ourier analysis as an algebraic theory {II}},
      ARXIV = {0911.1157},
     }
  • [szeg-3] B. Szegedy, Higher order Fourier analysis as an algebraic theory III.
    @misc{szeg-3,
      author={Szegedy, B.},
      TITLE={Higher order {F}ourier analysis as an algebraic theory {III}},
      ARXIV = {1001.4282},
     }
  • [structure] T. Tao, Structure and Randomness: Pages From Year One of a Mathematical Blog, Providence, RI: Amer. Math. Soc., 2008.
    @book {structure, MRKEY = {2459552},
      AUTHOR = {Tao, Terence},
      TITLE = {Structure and Randomness: Pages From Year One of a Mathematical Blog},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2008},
      PAGES = {xii+298},
      ISBN = {978-0-8218-4695-7},
      MRCLASS = {00-02},
      MRNUMBER = {2459552},
      MRREVIEWER = {W. T. Gowers},
      ZBLNUMBER = {05380664},
      }
  • [tao-ziegler] Go to document T. Tao and T. Ziegler, "The inverse conjecture for the Gowers norm over finite fields via the correspondence principle," Anal. PDE, vol. 3, iss. 1, pp. 1-20, 2010.
    @article {tao-ziegler, MRKEY = {2663409},
      AUTHOR = {Tao, Terence and Ziegler, Tamar},
      TITLE = {The inverse conjecture for the {G}owers norm over finite fields via the correspondence principle},
      JOURNAL = {Anal. PDE},
      FJOURNAL = {Analysis \& PDE},
      VOLUME = {3},
      YEAR = {2010},
      NUMBER = {1},
      PAGES = {1--20},
      ISSN = {1948-206X},
      MRCLASS = {11B30 (37A45)},
      MRNUMBER = {2663409},
      MRREVIEWER = {Andreas O. Bender},
      DOI = {10.2140/apde.2010.3.1},
      ZBLNUMBER = {05947646},
      }
  • [tao-vu] Go to document T. Tao and V. Vu, Additive Combinatorics, Cambridge: Cambridge Univ. Press, 2006, vol. 105.
    @book {tao-vu, MRKEY = {2289012},
      AUTHOR = {Tao, Terence and Vu, Van},
      TITLE = {Additive Combinatorics},
      SERIES = {Cambridge Stud. Adv. Math.},
      VOLUME = {105},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2006},
      PAGES = {xviii+512},
      ISBN = {978-0-521-85386-6; 0-521-85386-9},
      MRCLASS = {11-02 (05-02 05D10 11B13 11P70 11P82 28D05 37A45)},
      MRNUMBER = {2289012},
      MRREVIEWER = {Serge{\u\i} V. Konyagin},
      DOI = {10.1017/CBO9780511755149},
      ZBLNUMBER = {1127.11002},
      }
  • [john] Go to document T. Tao and V. Vu, "John-type theorems for generalized arithmetic progressions and iterated sumsets," Adv. Math., vol. 219, iss. 2, pp. 428-449, 2008.
    @article {john, MRKEY = {2435645},
      AUTHOR = {Tao, Terence and Vu, Van},
      TITLE = {John-type theorems for generalized arithmetic progressions and iterated sumsets},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {219},
      YEAR = {2008},
      NUMBER = {2},
      PAGES = {428--449},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {11B25},
      MRNUMBER = {2435645},
      MRREVIEWER = {Tom Sanders},
      DOI = {10.1016/j.aim.2008.05.002},
      ZBLNUMBER = {1165.11016},
      }
  • [vin] I. M. Vinogradov, The Method of Trigonometrical Sums in the Theory of Numbers, Mineola, NY: Dover Publications, 2004.
    @book {vin, MRKEY = {2104806},
      AUTHOR = {Vinogradov, I. M.},
      TITLE = {The Method of Trigonometrical Sums in the Theory of Numbers},
      NOTE = {translated from the Russian, revised and annotated by K. F. Roth and Anne Davenport},
      PUBLISHER = {Dover Publications},
      ADDRESS = {Mineola, NY},
      YEAR = {2004},
      PAGES = {x+180},
      ISBN = {0-486-43878-3},
      MRCLASS = {11Lxx (01A75)},
      MRNUMBER = {2104806},
      ZBLNUMBER = {1093.11001},
      }
  • [ziegler] Go to document T. Ziegler, "Universal characteristic factors and Furstenberg averages," J. Amer. Math. Soc., vol. 20, iss. 1, pp. 53-97, 2007.
    @article {ziegler, MRKEY = {2257397},
      AUTHOR = {Ziegler, Tamar},
      TITLE = {Universal characteristic factors and {F}urstenberg averages},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {20},
      YEAR = {2007},
      NUMBER = {1},
      PAGES = {53--97},
      ISSN = {0894-0347},
      MRCLASS = {37A30 (28D05 37A25)},
      MRNUMBER = {2257397},
      MRREVIEWER = {Randall McCutcheon},
      DOI = {10.1090/S0894-0347-06-00532-7},
      ZBLNUMBER = {1198.37014},
      }

Authors

Ben Green

DPMMS
Centre for Mathematical Sciences
University of Cambridge
Wilberforce Road
Cambridge CB3 0WB
England

Terence Tao

Department of Mathematics
University of California, Los Angeles
Los angeles, CA 90095-1555

Tamar Ziegler

Mathematics Department
Technion - Israel Institute of Technology
Haifa, 32000
Israel