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  2011, 18: 69-90. doi: 10.3934/era.2011.18.69

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

Ben Green 1, , Terence Tao 2, and  Tamar Ziegler 3,

1. 

Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA
2. 

Department of Mathematics, UCLA, 405 Hilgard Ave, Los Angeles, CA 90095
3. 

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

Received  March 2011 Revised  May 2011 Published  July 2011

This is an announcement of the proof of the inverse conjecture for the Gowers $U^{s+1}[N]$-norm for all $s \geq 3$; this is new for $s \geq 4$, the cases $s = 1,2,3$ having been previously established. More precisely we outline a proof that if $f : [N] \rightarrow [-1,1]$ is a function with ||$f$|| $U^{s+1}[N] \geq \delta$ then there is a bounded-complexity $s$-step nilsequence $F(g(n)\Gamma)$ which 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. In particular, one obtains an asymptotic formula for the number of $k$-term arithmetic progressions $p_1 < p_2 < ... < p_k \leq N$ of primes, for every $k \geq 3$.
Keywords: Gowers norms, nilsequences..
Mathematics Subject Classification: Primary: 11B9.
Citation: Ben Green, Terence Tao, Tamar Ziegler. An inverse theorem for the Gowers $U^{s+1}[N]$-norm. Electronic Research Announcements, 2011, 18: 69-90. doi: 10.3934/era.2011.18.69
References:
[1]

N. Alon, T. Kaufman, M. Krivelevich, S. Litsyn and D. Ron, Testing low-degree polynomials over GF(2),, Approximation, 51 (2005), 188.  doi: 10.1109/TIT.2005.856958.  Google Scholar

[2]

A. Balog and E. Szemerédi, A statistical theorem of set addition,, Combinatorica, 14 (1994), 263.  doi: 10.1007/BF01212974.  Google Scholar

[3]

V. Bergelson, B. Host and B. Kra, Multiple recurrence and nilsequences,, (with an appendix by I. Ruzsa), 160 (2005), 261.  doi: 10.1007/s00222-004-0428-6.  Google Scholar

[4]

V. Bergelson, T. Tao and T. Ziegler, An inverse theorem for uniformity seminorms associated with the action of $F_p^{\infty}$,, Geom. Funct. Anal., 19 (2010), 1539.  doi: 10.1007/s00039-010-0051-1.  Google Scholar

[5]

J.-P. Conze and E. Lesigne, Sur un théorème ergodique pour des mesures diagonales,, (French) [On an ergodic theorem for diagonal measures], 306 (1988), 491.   Google Scholar

[6]

N. Frantzikinakis, B. Host and B. Kra, Multiple recurrence and convergence for sequences related to the prime numbers,, J. Reine Angew. Math., 611 (2007), 131.  doi: 10.1515/CRELLE.2007.076.  Google Scholar

[7]

G. A. Freĭman, "Foundations of a Structural Theory of Set Addition,", Translations of Mathematical Monographs, 37 (1973).   Google Scholar

[8]

H. Furstenberg, "Nonconventional Ergodic Averages,", The legacy of John von Neumann (Hempstead, 50 (1988), 43.   Google Scholar

[9]

H. Furstenberg and B. Weiss, "A mean ergodic theorem for $1/N\sum^N_{n=1}f (T^n x)g(T^ {n^ 2}x)$,", Convergence in ergodic theory and probability (Columbus, 5 (1993), 193.   Google Scholar

[10]

W. T. Gowers, A new proof of Szemerédi's theorem for progressions of length four,, GAFA, 8 (1998), 529.  doi: 10.1007/s000390050065.  Google Scholar

[11]

-, A new proof of Szemerédi's theorem,, GAFA, 11 (2001), 465.   Google Scholar

[12]

B. Green, "Generalising the Hardy-Littlewood Method for Primes,", International Congress of Mathematicians. Vol. II, (2006), 373.   Google Scholar

[13]

B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions,, Annals of Math. (2), 167 (2008), 481.  doi: 10.4007/annals.2008.167.481.  Google Scholar

[14]

-, An inverse theorem for the Gowers $U^3$-norm, with applications,, Proc. Edinburgh Math. Soc., 51 (): 71.   Google Scholar

[15]

-, Linear equations in primes,, Ann. Math. (2), 171 (2010), 1753.   Google Scholar

[16]

-, The quantitative behaviour of polynomial orbits on nilmanifolds,, to appear in Ann. Math., ().   Google Scholar

[17]

-, The Möbius function is strongly orthogonal to nilsequences,, to appear in Ann. Math., ().   Google Scholar

[18]

-, An arithmetic regularity lemma, associated counting lemma, and applications,, in, 21 (): 261.   Google Scholar

[19]

B. Green, T. Tao and T. Ziegler, An inverse theorem for the Gowers $U^4[N]$-norm,, Glasgow Math. J., 53 (2011), 1.  doi: 10.1017/S0017089510000546.  Google Scholar

[20]

-, An inverse theorem for the Gowers $U^{s+1}[N]$ norm,, preprint, ().   Google Scholar

[21]

I. J. Håland, Uniform distribution of generalized polynomials,, J. Number Theory, 45 (1993), 327.  doi: 10.1006/jnth.1993.1082.  Google Scholar

[22]

B. Host and B. Kra, Convergence of Conze-Lesigne averages,, Erg. Th. Dyn. Sys., 21 (2001), 493.   Google Scholar

[23]

-, Averaging along cubes,, Modern Dynamical Systems and Applications, (2004), 123.   Google Scholar

[24]

-, Nonconventional ergodic averages and nilmanifolds,, Ann. of Math. (2), 161 (2005), 397.   Google Scholar

[25]

-, Uniformity seminorms on $l^\infty$ and applications,, J. Anal. Math., 108 (2009), 219.  doi: 10.1007/s11854-009-0024-1.  Google Scholar

[26]

A. Leibman, Pointwise convergence of ergodic averages of polynomial sequences of translations on a nilmanifold,, Ergodic Theory and Dynamical Systems, 25 (2005), 201.  doi: 10.1017/S0143385704000215.  Google Scholar

[27]

-, A canonical form and the distribution of values of generalized polynomials,, to appear in Israel Journal of Mathematics., ().   Google Scholar

[28]

E. Lesigne, Équations fonctionnelles, couplages de produits gauches et théorèmes ergodiques pour mesures diagonales,, (French) [Functional equations, 121 (1993), 315.   Google Scholar

[29]

I. Z. Ruzsa, Generalized arithmetical progressions and sumsets,, Acta Math. Hungar., 65 (1994), 379.  doi: 10.1007/BF01876039.  Google Scholar

[30]

A. Samorodnitsky, "Low-Degree Tests at Large Distances,", STOC'07, (2007), 506.   Google Scholar

[31]

B. Szegedy, Higher order Fourier analysis as an algebraic theory I,, preprint, ().   Google Scholar

[32]

-, Higher order Fourier analysis as an algebraic theory II,, preprint, ().   Google Scholar

[33]

-, Higher order Fourier analysis as an algebraic theory III,, preprint, ().   Google Scholar

[34]

T. Tao and T. Ziegler, The inverse conjecture for the Gowers norms over finite fields via the correspondence principle,, Anal. PDE, 3 (2010), 1.  doi: 10.2140/apde.2010.3.1.  Google Scholar

[35]

T. Tao and V. Vu, "Additive Combinatorics,", Cambridge Studies in Advanced Mathematics, 105 (2006).   Google Scholar

[36]

T. Ziegler, Universal characteristic factors and Furstenberg averages,, J. Amer. Math. Soc., 20 (2007), 53.  doi: 10.1090/S0894-0347-06-00532-7.  Google Scholar

show all references

References:
[1]

N. Alon, T. Kaufman, M. Krivelevich, S. Litsyn and D. Ron, Testing low-degree polynomials over GF(2),, Approximation, 51 (2005), 188.  doi: 10.1109/TIT.2005.856958.  Google Scholar

[2]

A. Balog and E. Szemerédi, A statistical theorem of set addition,, Combinatorica, 14 (1994), 263.  doi: 10.1007/BF01212974.  Google Scholar

[3]

V. Bergelson, B. Host and B. Kra, Multiple recurrence and nilsequences,, (with an appendix by I. Ruzsa), 160 (2005), 261.  doi: 10.1007/s00222-004-0428-6.  Google Scholar

[4]

V. Bergelson, T. Tao and T. Ziegler, An inverse theorem for uniformity seminorms associated with the action of $F_p^{\infty}$,, Geom. Funct. Anal., 19 (2010), 1539.  doi: 10.1007/s00039-010-0051-1.  Google Scholar

[5]

J.-P. Conze and E. Lesigne, Sur un théorème ergodique pour des mesures diagonales,, (French) [On an ergodic theorem for diagonal measures], 306 (1988), 491.   Google Scholar

[6]

N. Frantzikinakis, B. Host and B. Kra, Multiple recurrence and convergence for sequences related to the prime numbers,, J. Reine Angew. Math., 611 (2007), 131.  doi: 10.1515/CRELLE.2007.076.  Google Scholar

[7]

G. A. Freĭman, "Foundations of a Structural Theory of Set Addition,", Translations of Mathematical Monographs, 37 (1973).   Google Scholar

[8]

H. Furstenberg, "Nonconventional Ergodic Averages,", The legacy of John von Neumann (Hempstead, 50 (1988), 43.   Google Scholar

[9]

H. Furstenberg and B. Weiss, "A mean ergodic theorem for $1/N\sum^N_{n=1}f (T^n x)g(T^ {n^ 2}x)$,", Convergence in ergodic theory and probability (Columbus, 5 (1993), 193.   Google Scholar

[10]

W. T. Gowers, A new proof of Szemerédi's theorem for progressions of length four,, GAFA, 8 (1998), 529.  doi: 10.1007/s000390050065.  Google Scholar

[11]

-, A new proof of Szemerédi's theorem,, GAFA, 11 (2001), 465.   Google Scholar

[12]

B. Green, "Generalising the Hardy-Littlewood Method for Primes,", International Congress of Mathematicians. Vol. II, (2006), 373.   Google Scholar

[13]

B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions,, Annals of Math. (2), 167 (2008), 481.  doi: 10.4007/annals.2008.167.481.  Google Scholar

[14]

-, An inverse theorem for the Gowers $U^3$-norm, with applications,, Proc. Edinburgh Math. Soc., 51 (): 71.   Google Scholar

[15]

-, Linear equations in primes,, Ann. Math. (2), 171 (2010), 1753.   Google Scholar

[16]

-, The quantitative behaviour of polynomial orbits on nilmanifolds,, to appear in Ann. Math., ().   Google Scholar

[17]

-, The Möbius function is strongly orthogonal to nilsequences,, to appear in Ann. Math., ().   Google Scholar

[18]

-, An arithmetic regularity lemma, associated counting lemma, and applications,, in, 21 (): 261.   Google Scholar

[19]

B. Green, T. Tao and T. Ziegler, An inverse theorem for the Gowers $U^4[N]$-norm,, Glasgow Math. J., 53 (2011), 1.  doi: 10.1017/S0017089510000546.  Google Scholar

[20]

-, An inverse theorem for the Gowers $U^{s+1}[N]$ norm,, preprint, ().   Google Scholar

[21]

I. J. Håland, Uniform distribution of generalized polynomials,, J. Number Theory, 45 (1993), 327.  doi: 10.1006/jnth.1993.1082.  Google Scholar

[22]

B. Host and B. Kra, Convergence of Conze-Lesigne averages,, Erg. Th. Dyn. Sys., 21 (2001), 493.   Google Scholar

[23]

-, Averaging along cubes,, Modern Dynamical Systems and Applications, (2004), 123.   Google Scholar

[24]

-, Nonconventional ergodic averages and nilmanifolds,, Ann. of Math. (2), 161 (2005), 397.   Google Scholar

[25]

-, Uniformity seminorms on $l^\infty$ and applications,, J. Anal. Math., 108 (2009), 219.  doi: 10.1007/s11854-009-0024-1.  Google Scholar

[26]

A. Leibman, Pointwise convergence of ergodic averages of polynomial sequences of translations on a nilmanifold,, Ergodic Theory and Dynamical Systems, 25 (2005), 201.  doi: 10.1017/S0143385704000215.  Google Scholar

[27]

-, A canonical form and the distribution of values of generalized polynomials,, to appear in Israel Journal of Mathematics., ().   Google Scholar

[28]

E. Lesigne, Équations fonctionnelles, couplages de produits gauches et théorèmes ergodiques pour mesures diagonales,, (French) [Functional equations, 121 (1993), 315.   Google Scholar

[29]

I. Z. Ruzsa, Generalized arithmetical progressions and sumsets,, Acta Math. Hungar., 65 (1994), 379.  doi: 10.1007/BF01876039.  Google Scholar

[30]

A. Samorodnitsky, "Low-Degree Tests at Large Distances,", STOC'07, (2007), 506.   Google Scholar

[31]

B. Szegedy, Higher order Fourier analysis as an algebraic theory I,, preprint, ().   Google Scholar

[32]

-, Higher order Fourier analysis as an algebraic theory II,, preprint, ().   Google Scholar

[33]

-, Higher order Fourier analysis as an algebraic theory III,, preprint, ().   Google Scholar

[34]

T. Tao and T. Ziegler, The inverse conjecture for the Gowers norms over finite fields via the correspondence principle,, Anal. PDE, 3 (2010), 1.  doi: 10.2140/apde.2010.3.1.  Google Scholar

[35]

T. Tao and V. Vu, "Additive Combinatorics,", Cambridge Studies in Advanced Mathematics, 105 (2006).   Google Scholar

[36]

T. Ziegler, Universal characteristic factors and Furstenberg averages,, J. Amer. Math. Soc., 20 (2007), 53.  doi: 10.1090/S0894-0347-06-00532-7.  Google Scholar

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