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12
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].
The Möbius function is strongly orthogonal to nilsequences
 Ann. of Math
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THREE TOPICS IN ADDITIVE PRIME NUMBER THEORY
, 2007
"... 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 ..."
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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.
FROM HARMONIC ANALYSIS TO ARITHMETIC COMBINATORICS
, 2008
"... Arithmetic combinatorics, or additive combinatorics, is a fast developing area of research combining elements of number theory, combinatorics, harmonic analysis and ergodic theory. Its arguably bestknown result, and the one that brought it to global prominence, is the proof by Ben Green and Terence ..."
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Arithmetic combinatorics, or additive combinatorics, is a fast developing area of research combining elements of number theory, combinatorics, harmonic analysis and ergodic theory. Its arguably bestknown result, and the one that brought it to global prominence, is the proof by Ben Green and Terence Tao of the longstanding conjecture that primes contain arbitrarily long arithmetic progressions. There are many accounts and expositions of the GreenTao theorem, including the articles by Kra [119] and Tao [182] in the Bulletin. The purpose of the present article is to survey a broader, highly interconnected network of questions and results, built over the decades and spanning several areas of mathematics, of which the GreenTao theorem is a famous descendant. An old geometric problem lies at the heart of key conjectures in harmonic analysis. A major result in partial differential equations invokes combinatorial theorems on intersecting lines and circles. An unexpected argument points harmonic analysts towards additive number theory, with consequences that could have hardly been anticipated. We will not
1 A New Sifting function 1 ()nJ ω+ in
"... We define that prime equations 1 1 1 ( , ,), , ( ,)n k nf P P f P PL L L （5） are polynomials (with integer coefficients) irreducible over integers, where 1, , nP PL are all prime. If sifting function 1 ( ) 0nJ ω+ = then （5）has finite prime solutions. If 1 ( ) 0nJ ω+ ≠ then there are infin ..."
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We define that prime equations 1 1 1 ( , ,), , ( ,)n k nf P P f P PL L L （5） are polynomials (with integer coefficients) irreducible over integers, where 1, , nP PL are all prime. If sifting function 1 ( ) 0nJ ω+ = then （5）has finite prime solutions. If 1 ( ) 0nJ ω+ ≠ then there are infinitely many primes 1, , nP PL such that 1, kf fL are primes. We obtain a unite prime formula in prime distribution primes}are,,:,,{)1, ( 111 kffNPPnN knk LL ≤=++π
FROM HARMONIC ANALYSIS TO ARITHMETIC COMBINATORICS: A BRIEF SURVEY
"... The purpose of this note is to showcase a certain line of research that connects harmonic analysis, specifically restriction theory, to other areas of mathematics such as PDE, geometric measure theory, combinatorics, and number theory. There are many excellent indepth presentations of the various ..."
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The purpose of this note is to showcase a certain line of research that connects harmonic analysis, specifically restriction theory, to other areas of mathematics such as PDE, geometric measure theory, combinatorics, and number theory. There are many excellent indepth presentations of the various areas of research that we will discuss; see e.g., the references below. The emphasis here will be on highlighting the connections between these areas. Our starting point will be restriction theory in harmonic analysis on Euclidean spaces. The main theme of restriction theory, in this context, is the connection between the decay at infinity of the Fourier transforms of singular measures and the geometric properties of their support, including (but not necessarily limited to) curvature and dimensionality. For example, the Fourier transform of a measure supported on a hypersurface in Rd need not, in general, belong to any Lp with p < ∞, but there are positive results if the hypersurface in question is curved. A classic example is the restriction theory for the sphere, where a conjecture due to E. M. Stein asserts that the