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 subnilmanifolds 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 …

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+ …