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14
The quantitative behaviour of polynomial orbits on nilmanifolds
, 2007
"... A theorem of Leibman [19] 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 o ..."
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Cited by 38 (3 self)
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A theorem of Leibman [19] 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 factorization 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 subsequent paper [13] we shall use this theorem to establish the Möbius and Nilsequences conjecture from our earlier paper [12].
Generalising the HardyLittlewood method for primes
 IN: PROCEEDINGS OF THE INTERNATIONAL CONGRESS OF MATHEMATICIANS
, 2007
"... The HardyLittlewood method is a wellknown technique in analytic number theory. Among its spectacular applications are Vinogradov’s 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving an asymptotic for the number o ..."
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Cited by 12 (6 self)
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The HardyLittlewood method is a wellknown technique in analytic number theory. Among its spectacular applications are Vinogradov’s 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving an asymptotic for the number of 3term progressions of primes, all less than N. This article surveys recent developments of the author and T. Tao, in which the HardyLittlewood method has been generalised to obtain, for example, an asymptotic for the number of 4term arithmetic progressions of primes less than N.
Nonconventional ergodic averages and multiple recurrence for von Neumann dynamical systems
 Pacific J. Math
"... Abstract. The Furstenberg recurrence theorem (or equivalently, Szemerédi’s theorem) can be formulated in the language of von Neumann algebras as follows: given an integer k ≥ 2, an abelian finite von Neumann algebra (M, τ) with an automorphism α: M → M, and a nonnegative a ∈ M with τ(a)> 0, on ..."
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Abstract. The Furstenberg recurrence theorem (or equivalently, Szemerédi’s theorem) can be formulated in the language of von Neumann algebras as follows: given an integer k ≥ 2, an abelian finite von Neumann algebra (M, τ) with an automorphism α: M → M, and a nonnegative a ∈ M with τ(a)> 0, one has lim infN→ ∞ 1N ∑N n=1 Re τ(aα n(a)... α(k−1)n(a))> 0; a subsequent result of Host and Kra shows that this limit exists. In particular, Re τ(aαn(a)... α(k−1)n(a))> 0 for all n in a set of positive density. From the von Neumann algebra perspective, it is thus natural to ask to what extent these results remain true when the abelian hypothesis is dropped. All three claims hold for k = 2, and we show in this paper that all three claims hold for all k when the von Neumann algebra is asymptotically abelian, and that the last two claims hold for k = 3 when the von Neumann algebra is
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"... On the structure of steps of threeterm arithmetic progressions in a dense set of integers ..."
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On the structure of steps of threeterm arithmetic progressions in a dense set of integers