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**The quantitative behaviour of polynomial orbits on nilmanifolds.**
*(English)*
Zbl 1251.37012

Ann. Math. (2) 175, No. 2, 465-540 (2012); erratum ibid. 179, No. 3, 1175-1183 (2014).

This paper provides the new quantitative results for dynamics on nilmanifolds needed for the resolution of the Möbius and nilsequence conjectures from the authors’ work on linear patterns in primes [Ann. Math. (2) 171, No. 3, 1753–1850 (2010; Zbl 1242.11071)]. These quantitative results are of great indepedent interest in dynamics. While the developments are of necessity both complicated and technical, the exposition is extremely careful and there is a careful motivation via earlier results. The starting point is equidistribution for linear sequences in the following sense. A linear expression \((n\alpha +\beta)\) in the additive circle either takes on only finitely many values or is equidistributed in the circle. This dichotomy echoes through much of the background, and roughly speaking the earlier work develops this idea to more complex types of sequences (replacing the linear expression by a polynomial) on more complex spaces, namely nilmanifolds (in the non-abelian setting ‘polynomial’ means vanishing under finitely many applications of a formal differentiation operator sending a function \(g\) to the function \(\delta g\) defined by \((\delta(g))(n)=g(n+1)g(n)^{-1}\)). The history is traced carefully, describing results of Kronecker, Weyl, (Leon) Green and Parry, and reaching the result of A. Leibman [Ergodic Theory Dyn. Syst. 25, No. 1, 201–213 (2005; Zbl 1080.37003)] which gives a similar dichotomy in the setting of polynomials on nilmanifolds. For a polynomial sequence \(g:\mathbb Z\rightarrow G\) and a nilmanifold \(G/\Gamma\), Leibman shows that exactly one of the following statements holds: the sequence \((g(n)\Gamma)\) is equidistributed in \(G/\Gamma\), or there is a non-trivial character \(\eta:G\rightarrow\mathbb R/\mathbb Z\) such that \(\eta\circ g\) is constant. This is itself a major generalisation of Weyl’s polynomial equidistribution theorem.

In this paper a third and highly significant generalization is added to the step from linear to polynomial expressions and the step from the circle to a nilmanifold, and that is to render quantitative the statement of Leibman’s theorem. In this development a quantitative form of Weyl’s polynomial equidistribution is proved (using exponential sum bounds) and used, along with a great many other quantitative improvements of known results. The final result (in the same setting as the result of Leibman) gives precise expression to the statement that the initial \(N\) terms of the sequence \((g(n)\Gamma)\) is either \(\delta\)-close to equidistributed, or is very far from being equidistributed up to time \(\delta^{O_{m,d}(1)}N\) in that it is very close to a union of \(\delta^{-O_{m,d}(1)}\) subtori (here \(m\) is the topological dimension of the nilmanifold and \(d\) is a group-theoretic parameter associated to a filtration of \(G\)). Despite the length and technical nature of the work, the results both intermediate and final are of great interest, and the exposition is exceptionally careful and readable.

In this paper a third and highly significant generalization is added to the step from linear to polynomial expressions and the step from the circle to a nilmanifold, and that is to render quantitative the statement of Leibman’s theorem. In this development a quantitative form of Weyl’s polynomial equidistribution is proved (using exponential sum bounds) and used, along with a great many other quantitative improvements of known results. The final result (in the same setting as the result of Leibman) gives precise expression to the statement that the initial \(N\) terms of the sequence \((g(n)\Gamma)\) is either \(\delta\)-close to equidistributed, or is very far from being equidistributed up to time \(\delta^{O_{m,d}(1)}N\) in that it is very close to a union of \(\delta^{-O_{m,d}(1)}\) subtori (here \(m\) is the topological dimension of the nilmanifold and \(d\) is a group-theoretic parameter associated to a filtration of \(G\)). Despite the length and technical nature of the work, the results both intermediate and final are of great interest, and the exposition is exceptionally careful and readable.

Reviewer: Thomas B. Ward (Durham)

### MSC:

37A45 | Relations of ergodic theory with number theory and harmonic analysis (MSC2010) |

11N13 | Primes in congruence classes |

### Keywords:

quantitative equidistribution; nilmanifold; polynomial sequence; Möbius function; randomness
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\textit{B. Green} and \textit{T. Tao}, Ann. Math. (2) 175, No. 2, 465--540 (2012; Zbl 1251.37012)

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