The quantitative behaviour of polynomial orbits on nilmanifolds

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 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 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.

The Möbius function is strongly orthogonal to nilsequences

“…Recall from [7,Def. 1.2(iv)] that a sequence (g(n)Γ) n∈ [N ] in a nilmanifold is totally δ-equidistributed if we have In the next section we shall establish the following result about the lack of correlation of Möbius with equidistributed nilsequences.…”

“…We start with a brief overview. The main ingredient of this argument is [7,Th. 1.19], that is to say the factorisation g = εg γ mentioned above.…”

“…Finally, a fair amount of what might be called "quantitative nil-linear algebra" is required to keep track of the various nilmanifolds and Lipschitz functions involved in the argument. Here we draw repeatedly on the material assembled in [7,Appendix A] for this purpose; we encourage the reader to gloss over these essentially routine issues on a first reading.…”

“…This paper is intimately tied to, and is intended to be read in conjunction with, the longer companion paper [7], which proves results about the distribution of finite polynomial orbits on nilmanifolds. In particular, we shall make heavy use of the notation and lemmas from that paper.…”

Multifractal behavior of polynomial Fourier series

“…In this interval we seek cancellation caused by the oscillation of e aQ(n)/p ν F −m . By Proposition 4.3 of [GT12], we have that there exists K > 0 such that for any 0 < δ < 1/2 we have (2.12)…”