×

zbMATH — the first resource for mathematics

Quadratic uniformity of the Möbius function. (English) Zbl 1160.11017
Since proving their celebrated theorem on arithmetic progressions in the primes [Ann. Math. (2) 171, No. 3, 1753–1850 (2010; Zbl 1242.11071)] the authors have embarked on a program to handle the solution of a wide class of systems of linear equations in the primes. The program has two components, the so-called Gowers inverse conjectures, and the Möbius nilsequence conjectures, and the paper under review establishes the first new case of the latter.
The motivation behind the paper is the metaprinciple that the Möbius function behaves sufficiently randomly that it is asymptotically orthogonal to any low complexity object. Our classical knowledge in this regard may be reformulated in the language of the paper as the Möbius nilsequence conjecture (theorem) for 1-step nilsequences, which we record now from Example 4 of the paper.
Suppose that \(G\) is a connected, simply-connected abelian Lie group with a smooth metric \(d\), and that \(\Gamma\) is a closed cocompact subgroup of \(G\). Then \(G/\Gamma\) is called a \(1\)-step nilmanifold; it is a torus. Furthermore suppose that \(F:G/\Gamma \rightarrow \mathbb C\) is a Lipschitz function of constant \(1\), and write \(T_g:G/\Gamma \rightarrow G/\Gamma\) for left translation by \(g\) on \(G/\Gamma\). Then one has the estimate \[ \mathbf E_{n \in [N]}{\mu(n)\overline{F(T_g^nx)}} = O_{A,G/\Gamma}(\log^{-A}N) \] for all \(A>0\), integers \(N\geq 1\), elements \(g \in G\) and \(x \in G/\Gamma\). In the paper the authors establish this in the more general case when \(G/\Gamma\) is a \(2\)-step nilmanifold. The proof is long but written in a very accessible style and many of the ingredients will be recognized by those having some familiarity with analytic number theory. Comprehensive appendices make the paper essentially self-contained.

MSC:
11B99 Sequences and sets
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] Auslander, L.; Green, L.; Hahn, F., Flows on homogeneous spaces, (1963), Princeton University Press, Princeton, N.J. · Zbl 0106.36802
[2] Bilu, Yuri, Structure of sets with small sumset, Astérisque, 258, xi, 77-108, (1999) · Zbl 0946.11004
[3] Bourbaki, Nicolas, Lie groups and Lie algebras. Chapters 1-3, (1998), Springer-Verlag, Berlin · Zbl 0672.22001
[4] Bourgain, J., On \(Λ (p)\)-subsets of squares, Israel J. Math., 67, 3, 291-311, (1989) · Zbl 0692.43005
[5] Corwin, Lawrence J.; Greenleaf, Frederick P., Representations of nilpotent Lie groups and their applications. Part I, 18, (1990), Cambridge University Press, Cambridge · Zbl 0704.22007
[6] Davenport, H., On some infinite series involving arithmetical functions. II, Quart. J. Math. Oxf., 8, 313-320, (1937) · Zbl 0017.39101
[7] Davenport, Harold, Multiplicative number theory, 74, (2000), Springer-Verlag, New York · Zbl 1002.11001
[8] Furstenberg, Hillel, The legacy of John von Neumann (Hempstead, NY, 1988), 50, Nonconventional ergodic averages, 43-56, (1990), Amer. Math. Soc., Providence, RI · Zbl 0711.28006
[9] Gowers, W. T., A new proof of szemerédi’s theorem, Geom. Funct. Anal., 11, 3, 465-588, (2001) · Zbl 1028.11005
[10] Green, B. J.; Tao, T. C., Linear equations in primes
[11] Green, B. J.; Tao, T. C., An inverse theorem for the Gowers \(U^3\)-norm, Proc. Edinburgh Math. Soc., 51, 1, 73-153, (2008) · Zbl 1202.11013
[12] Green, B. J.; Tao, T. C., The primes contain arbitrarily long arithmetic progressions, Annals of Math., 167, 481-547, (2008) · Zbl 1191.11025
[13] Green, Ben, Surveys in combinatorics 2005, 327, Finite field models in additive combinatorics, 1-27, (2005), Cambridge Univ. Press, Cambridge · Zbl 1155.11306
[14] Hua, L. K., Some results in the additive prime number theory, Quart. J. Math. Oxford, 9, 68-80, (1938) · JFM 64.0131.02
[15] Iwaniec, Henryk; Kowalski, Emmanuel, Analytic number theory, 53, (2004), American Mathematical Society, Providence, RI · Zbl 1059.11001
[16] Mal’cev, A. I., On a class of homogeneous spaces, Izvestiya Akad. Nauk. SSSR. Ser. Mat., 13, 9-32, (1949)
[17] Montgomery, Hugh L., Ten lectures on the interface between analytic number theory and harmonic analysis, 84, (1994), Published for the Conference Board of the Mathematical Sciences, Washington, DC · Zbl 0814.11001
[18] Ruzsa, Imre Z., On an additive property of squares and primes, Acta Arith., 49, 3, 281-289, (1988) · Zbl 0636.10042
[19] Tao, Terence, Arithmetic progressions and the primes, Collect. Math., Vol. Extra, 37-88, (2006) · Zbl 1109.11043
[20] Tao, Terence; Vu, Van, Additive combinatorics, 105, (2006), Cambridge University Press, Cambridge · Zbl 1127.11002
[21] Vaughan, R. C., The Hardy-Littlewood method, 125, (1997), Cambridge University Press, Cambridge · Zbl 0868.11046
[22] Vaughan, Robert-C., Sommes trigonométriques sur LES nombres premiers, C. R. Acad. Sci. Paris Sér. A-B, 285, 16, A981-A983, (1977) · Zbl 0374.10025
[23] Vinogradov, I. M., Some theorems concerning the primes, Mat. Sbornik. N.S., 2, 179-195, (1937) · Zbl 0017.19803
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.