The quantitative behaviour of polynomial orbits on nilmanifolds

By Ben Joseph Green, Terence Tao. 2007

252 citations recorded by the source corpus.

A theorem of Leibman asserts that a polynomial orbit $(g(1),g(2),g(3),\ldots)$ on a nilmanifold $G/\Gamma$ is always equidistributed in a union of closed sub-nilmanifolds of $G/\Gamma$. In this paper we give a quantitative version of Leibman's result, describing the uniform distribution properties of a finite polynomial orbit $(g(1),\ldots,g(N))$ in a nilmanifold. More specifically we show that there is a factorization $g = \epsilon g'\gamma$, where $\epsilon(n)$ is "smooth", $\gamma(n)$ is periodic and "rational", and $(g'(a),g'(a+d),\ldots,g'(a + d(l-1)))$ is uniformly distributed (up to a specified error $\delta$) inside some subnilmanifold $G'/\Gamma'$ of $G/\Gamma$, for all sufficiently dense arithmetic progressions $a,a+d,\ldots,a+d(l-1)$ inside $\{1,..,N\}$. Our bounds are uniform in $N$ and are polynomial in the error tolerance delta. In a subsequent paper we shall use this theorem to establish the Mobius and Nilsequences conjecture from our earlier paper "Linear equations in primes".

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d479c385b9dad44c778a08908a7e4d3804230e07

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

The logarithmic Sarnak conjecture for ergodic weights (2017, unknown): “We give a sketch of the proof of a more general identity in Appendix C; the argument is almost entirely based on the argument given by Tao in [65].”
Higher order Fourier analysis of multiplicative functions and applications (2014, methodology): “6In [23] a function with this property is said to have vertical oscillation h.”
Some open problems on multiple ergodic averages (2011, background): “such equidistribution problems have been developed in recent years, thus making such a reduction very much worthwhile. Some examples of useful equidistribution results on nilmanifolds can be found in [17, 130, 131, 141, 169, 170, 171, 185, 227, 228, 230, 231, 233, 241, 242, 299]. 2.6. A general stra”
Nil Bohr-sets and almost automorphy of higher order (2014, background): “[31] B. Green, T. Tao and T. Ziegler, An inverse theorem for the Gowers Us+1[N ] norm, Ann. of Math., 176 (2012), no. 2, 1231–1372.”
Long gaps between primes (2014, methodology): “In [12], the approach taken was to use recent results on line ar equations in primes [19, 18, 20] to locate arithmetic progre ssionsq, q + r!p, .”
Multiple recurrence and convergence for hardy sequences of polynomial growth (2009, background): “The proofs of all the results mentioned below can be found or deduced from [28] and [22].”
ON THE DEVIATION OF ERGODIC AVERAGES FOR HOROCYCLE FLOWS (2013, result): “Important results on effective equidistribution in other homogeneous spaces have been obtained by Einsiedler, Margulis and Venkatesh [9], and by Green and Tao [12].”