The quantitative behaviour of polynomial orbits on nilmanifolds

2007/09/22 by Green, Ben, Tao, Terence · 8 citations
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.0709.3562

Abstract

A theorem of Leibman asserts that a polynomial orbit (g(1),g(2),g(3),…) 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(1),…,g(N)) in a nilmanifold. More specifically we show that there is a factorization g = εg'γ, where ε(n) is "smooth", γ(n) is periodic and "rational", and (g'(a),g'(a+d),…,g'(a + d(l-1))) is uniformly distributed (up to a specified error δ) inside some subnilmanifold G'/Γ' of G/Γ, for all sufficiently dense arithmetic progressions a,a+d,…,a+d(l-1) inside \1,..,N\. Our bounds are uniform in N and are polynomial in the error tolerance delta. In a subsequent paper we shall use this theorem to establish the Mobius and Nilsequences conjecture from our earlier paper "Linear equations in primes".

Cited by

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• An inverse theorem for all finite abelian groups via nilmanifolds
• The distribution of prime values of random polynomials
• New Obstacles to Multiple Recurrence
• Almost Countable Spectrum and Logarithmic Sarnak Conjecture
• Pointwise convergence of ergodic averages along quadratic bracket polynomials
• Density of orbits of horocycle flows at sub-quadratic polynomial times
• Finding arithmetic progressions in dense sets of integers

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