The quantitative behaviour of polynomial orbits on nilmanifolds

@article{Green2007TheQB,
  title={The quantitative behaviour of polynomial orbits on nilmanifolds},
  author={Ben Green and Terence Tao},
  journal={arXiv: Number Theory},
  year={2007}
}
  • B. Green, T. Tao
  • Published 22 September 2007
  • Mathematics
  • arXiv: Number Theory
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… 

Figures from this paper

Uniform distribution in nilmanifolds along functions from a Hardy field
We study equidistribution properties of translations on nilmanifolds along functions of polynomial growth from a Hardy field. More precisely, if $X=G/\Gamma$ is a nilmanifold, $a_1,\ldots,a_k\in G$
The Mobius function is strongly orthogonal to nilsequences
We show that the Mobius function (n) is strongly asymptotically or- thogonal to any polynomial nilsequence (F (g(n))) n2N. Here, G is a sim- ply-connected nilpotent Lie group with a discrete and
On the Liouville function at polynomial arguments
Let $\lambda$ denote the Liouville function. A problem posed by Chowla and by Cassaigne-Ferenczi-Mauduit-Rivat-Sarkozy asks to show that if $P(x)\in \mathbb{Z}[x]$, then the sequence $\lambda(P(n))$
Higher order almost automorphy, recurrence sets and the regionally proximal relation
In this paper, $d$-step almost automorphic systems are studied for $d\in\N$, which are the generalization of the classical almost automorphic ones. For a minimal topological dynamical system $(X,T)$
Sarnak's Conjecture for nilsequences on arbitrary number fields and applications.
We formulate the generalized Sarnak's Mobius disjointness conjecture for an arbitrary number field $K$, and prove a quantitative disjointness result between polynomial nilsequences
Convergence Results for Systems of Linear Forms on Cyclic Groups and Periodic Nilsequences
TLDR
An extension of Croot's theorem, showing that the result holds for k-term progressions for general $k$ and further for all systems of integer linear forms of finite complexity, and for the maximum densities of sets free of solutions to systems of linear equations.
Probabilistic nilpotence in infinite groups
The 'degree of k-step nilpotence' of a finite group G is the proportion of the tuples (x_1,...,x_{k+1}) in G^{k+1} for which the simple commutator [x_1,...,x_{k+1}] is equal to the identity. In this
Equidistribution of sparse sequences on nilmanifolds
AbstractWe study equidistribution properties of nil-orbits (bnx)n∈ℕ when the parameter n is restricted to the range of some sparse sequence that is not necessarily polynomial. For example, we show
Möbius disjointness for nilsequences along short intervals
For a nilmanifold $G/\Gamma$, a $1$-Lipschitz continuous function $F$ and the Mobius sequence $\mu(n)$, we prove a bound on the decay of the averaged short interval correlation
Coherence measures for heralded single-photon sources
We show that the Mobius function mu(n) is strongly asymptotically orthogonal to any polynomial nilsequence n -> F(g(n)L). Here, G is a simply-connected nilpotent Lie group with a discrete and
...
...

References

SHOWING 1-10 OF 59 REFERENCES
Multiple recurrence and nilsequences
Aiming at a simultaneous extension of Khintchine’s and Furstenberg’s Recurrence theorems, we address the question if for a measure preserving system $(X,\mathcal{X},\mu,T)$ and a set
Spectra of nilflows
Recently L. Auslander, F. Hahn, and L. Markus [ l ] have examined an important class of dynamical systems, proving the existence of minimal, distal, nonequicontinuous flows on compact nilmanifolds.
Universal characteristic factors and Furstenberg averages
Let X=(X^0,\mu,T) be an ergodic measure preserving system. For a natural number k we consider the averages (*) 1/N \sum_{n=1}^N \prod_{j=1}^k f_j(T^{n a_j}x) where the functions f_j are bounded, and
Pointwise convergence of ergodic averages for polynomial actions of $\mathbb{Z}^{d}$ by translations on a nilmanifold
  • A. Leibman
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 2004
Generalizing the one-parameter case, we prove that the orbit of a point on a compact nilmanifold X under a polynomial action of $\mathbb{Z}^{d}$ by translations on X is uniformly distributed on the
Ratner's Theorems on Unipotent Flows
The theorems of Berkeley mathematician, Marina Ratner have guided key advances in the understanding of dynamical systems. Unipotent flows are well-behaved dynamical systems, and Ratner has shown that
Linear equations in primes
Consider a system ψ of nonconstant affine-linear forms ψ 1 , ... , ψ t : ℤ d → ℤ, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [-N, N] d be convex. A generalisation
Invariant Measures and Orbit Closures on Homogeneous Spaces for Actions of Subgroups Generated by Unipotent Elements
The theorems of M. Ratner, describing the finite ergodic invariant measures and the orbit closures for unipotent flows on homogeneous spaces of Lie groups, are extended for actions of subgroups
Uniformity seminorms on ℓ∞ and applications
A key tool in recent advances in understanding arithmetic progressions and other patterns in subsets of the integers is certain norms or seminorms. One example is the norms on ℤ/Nℤ introduced by
Nonconventional ergodic averages and nilmanifolds
We study the L2-convergence of two types of ergodic averages. The first is the average of a product of functions evaluated at return times along arithmetic progressions, such as the expressions
Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold
  • A. Leibman
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 2004
We show that the orbit of a point on a compact nilmanifold X under the action of a polynomial sequence of translations on X is well distributed on the union of several sub-nilmanifolds of X. This
...
...