Resumen de The quantitative behaviour of polynomial orbits on nilmanifolds

Benjamin Green, Terence Tao

• A theorem of Leibman asserts that a polynomial orbit (g(n)G) n?Z on a nilmanifold G/G is always equidistributed in a union of closed sub-nilmanifolds of G/G . In this paper we give a quantitative version of Leibman�s result, describing the uniform distribution properties of a finite polynomial orbit (g(n)G) n?[N] in a nilmanifold. More specifically we show that there is a factorisation g=eg ' ? , where e(n) is �smooth,� (?(n)G) n?Z is periodic and �rational,� and (g ' (n)G) n?P is uniformly distributed (up to a specified error d ) inside some subnilmanifold G ' /G ' of G/G for all sufficiently dense arithmetic progressions P?[N] .

  Our bounds are uniform in N and are polynomial in the error tolerance d . In a companion paper we shall use this theorem to establish the Möbius and Nilsequences conjecture from an earlier paper of ours.