# The quantitative behaviour of polynomial orbits on nilmanifolds | Annals of Mathematics

Abstract A theorem of Leibman asserts that a polynomial orbit (g(n)Γ)n∈ℤ on a nilmanifold G/Γ is always equidistributed in a union of closed sub-nilmanifolds of G/Γ. In this paper we give a quantitative version of Leibman’s result, describing the uniform distribution properties of a finite polynomial orbit (g(n)Γ)n∈[N] in a nilmanifold. More specifically we show that there is a factorisation g=εg′γ, where ε(n) is “smooth,” (γ(n)Γ)n∈ℤ is periodic and “rational,” and (g′(n)Γ)n∈P is uniformly distributed (up to a specified error δ) inside some subnilmanifold G′/Γ′ of G/Γ for all sufficiently dense arithmetic progressions P⊆[N]. Our bounds are uniform in N and are polynomial in the error tolerance δ. In a companion paper we shall use this theorem to establish the Möbius and Nilsequences conjecture from an earlier paper of ours.

KEYWORDS

SHARE & LIKE

### 数学年刊（Annals of Mathematics）

0 Following 0 Fans 0 Projects 674 Articles

SIMILAR ARTICLES

Abstract For any nondegenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a correspondin

Abstract For a large class of nonlinear Schrödinger equations with nonzero conditions at infinity and for any speed c less than the sound velocity, we

Abstract Let L2,p(ℝ2) be the Sobolev space of real-valued functions on the plane whose Hessian belongs to Lp. For any finite subset E⊂ℝ2 and p>2, let

Abstract We prove that for any group G in a fairly large class of generalized wreath product groups, the associated von Neumann algebra LG completely

Abstract We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We

Abstract This paper has two main results. Firstly, we complete the parametrisation of all p-blocks of finite quasi-simple groups by finding the so-cal

Abstract We derive sharp Moser-Trudinger inequalities on the CR sphere. The first type is in the Adams form, for powers of the sublaplacian and for ge

Abstract We prove that isoparametric hypersurfaces with (g,m)=(6,2) are homogeneous, which answers Dorfmeister-Neher’s conjecture affirmatively and so

Abstract We prove the periodicity conjecture for pairs of Dynkin diagrams using Fomin-Zelevinsky’s cluster algebras and their (additive) categorificat

Abstract If F(x,y)∈ℤ[x,y] is an irreducible binary form of degree k≥3, then a theorem of Darmon and Granville implies that the generalized superellipt