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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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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].
FROM HARMONIC ANALYSIS TO ARITHMETIC COMBINATORICS
, 2006
"... We will describe a certain line of research connecting classical harmonic analysis to PDE regularity estimates, an old question in Euclidean geometry, a variety of deep combinatorial problems, recent advances in analytic number theory, and more. Traditionally, restriction theory is a part of classi ..."
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We will describe a certain line of research connecting classical harmonic analysis to PDE regularity estimates, an old question in Euclidean geometry, a variety of deep combinatorial problems, recent advances in analytic number theory, and more. Traditionally, restriction theory is a part of classical Fourier analysis that investigates the relationship between geometric and Fourieranalytic properties of singular measures. It became clear over the years that the theory would have to involve sophisticated geometric and combinatorial input. Two particularly important turning points were Fefferman’s work in the 1970s invoking the ”Kakeya problem” in this context, and Bourgain’s application of Gowers’s additive number theory techniques to the Kakeya problem almost 30 years later. All this led harmonic analysts to explore areas previously foreign to them, such as combinatorial geometry, graph theory, and additive number theory. Although the Kakeya and restriction problems remain stubbornly open, the
THREE TOPICS IN ADDITIVE PRIME NUMBER THEORY
, 710
"... Abstract. We discuss, in varying degrees of detail, three contemporary themes in prime number theory. Topic 1: the work of Goldston, Pintz and Yıldırım on short gaps between primes. Topic 2: the work of Mauduit and Rivat, establishing that 50% of the primes have odd digit sum in base 2. Topic 3: wor ..."
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Abstract. We discuss, in varying degrees of detail, three contemporary themes in prime number theory. Topic 1: the work of Goldston, Pintz and Yıldırım on short gaps between primes. Topic 2: the work of Mauduit and Rivat, establishing that 50% of the primes have odd digit sum in base 2. Topic 3: work of Tao and the author on linear equations in primes. Introduction. These notes are to accompany two lectures I am scheduled to give at the Current Developments in Mathematics conference at Harvard in November 2007. The title of those lectures is ‘A good new millennium for primes’, but I have chosen a rather drier title for these notes for two reasons. Firstly, the title of the lectures was unashamedly stolen (albeit with permission) from Andrew Granville’s entertaining