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DECOMPOSITIONS, APPROXIMATE STRUCTURE, TRANSFERENCE, AND THE HAHNBANACH THEOREM
, 2008
"... We discuss three major classes of theorems in additive and extremal combinatorics: decomposition theorems, approximate structure theorems, and transference principles. We also show how the finitedimensional HahnBanach theorem can be used to give short and transparent proofs of many results of the ..."
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We discuss three major classes of theorems in additive and extremal combinatorics: decomposition theorems, approximate structure theorems, and transference principles. We also show how the finitedimensional HahnBanach theorem can be used to give short and transparent proofs of many results of these kinds. Amongst the applications of this method is a much shorter proof of one of the major steps in the proof of Green and Tao that the primes contain arbitrarily long arithmetic progressions. In order to explain the role of this step, we include a brief description of the rest of their argument. A similar proof has been discovered independently by Reingold, Trevisan, Tulsiani and Vadhan [RTTV].
Arithmetic progressions and the primes  El Escorial lectures
 Collectanea Mathematica (2006), Vol. Extra., 3788 (Proceedings of the 7th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial
"... Abstract. We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length k in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes. 1. ..."
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Abstract. We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length k in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes. 1.
GREENTAO THEOREM IN FUNCTION FIELDS
, 2009
"... We adapt the proof of the GreenTao theorem on arithmetic progressions in primes to the setting of polynomials over a finite field, to show that for every k, the irreducible polynomials in Fq[t] contain configurations of the form {f + Pg: deg(P) < k}, g ̸ = 0. ..."
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We adapt the proof of the GreenTao theorem on arithmetic progressions in primes to the setting of polynomials over a finite field, to show that for every k, the irreducible polynomials in Fq[t] contain configurations of the form {f + Pg: deg(P) < k}, g ̸ = 0.
A NOTE ON TWIN PRIMES
, 2005
"... ABSTRACT. We relate the twin prime conjecture to corresponding conjectures for a short divisor sum which approximates primes. The twin prime conjecture states that there are infinitely many pairs of primes differing by two. More generally we expect there will be infinitely many pairs of primes with ..."
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ABSTRACT. We relate the twin prime conjecture to corresponding conjectures for a short divisor sum which approximates primes. The twin prime conjecture states that there are infinitely many pairs of primes differing by two. More generally we expect there will be infinitely many pairs of primes with difference k, for any fixed even integer k. Let ƒ.n / be the von Mangoldt function defined to be log p if n D pm, m 1, and zero otherwise. Then a quantitative version of the general twin prime conjecture is that, as N!1, NX (1) ƒ.n/ƒ.n C k / D.S.k / C o.1//N; nD1 where S.k / is the singular series given by 8
Prvosla Obsahuj Libovoln Dlouh Aritmetick Posloupnosti
"... ektivn, dv konkrtn funkci f : N ! N takovou, e mnoina f1; 2; : : : ; f(k)g pro kad k obsahuje aritmetickou posloupnost dlky k sloenou z prvosel. Tao v [49] uvd, e lze vzt f(k) = 2 2 2 2 2 100k Green a Tao modi kac dkazu vty 1.1 dokzali jej zeslen, vtu 1.2: Jeli P mnoina vech prvosel a podmn ..."
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ektivn, dv konkrtn funkci f : N ! N takovou, e mnoina f1; 2; : : : ; f(k)g pro kad k obsahuje aritmetickou posloupnost dlky k sloenou z prvosel. Tao v [49] uvd, e lze vzt f(k) = 2 2 2 2 2 100k Green a Tao modi kac dkazu vty 1.1 dokzali jej zeslen, vtu 1.2: Jeli P mnoina vech prvosel a podmnoina Q P spluje lim sup n!1 Q(n) P (n) = c > 0 (Q(n) je poet prvk q v Q, q n, podobn P (n)), mus Q obsahovat libovoln dlouh aritmetick posloupnosti. Je znmo, e pro Q 1 = fp 2 P : p = 4n+1g mme c = 1=2 a kad p 2 Q 1 je souet dvou tverc (p = a +b pro dv pirozen sla a; b, viz st 3). Vta 1.2 tedy dv (napklad) dosud neznm fakt, e existuj libovoln dlouh aritmetick posloupnosti tvoen souty dvou tverc. (Nap. 37 = 1 , 61 = 5 , 85 = 9 + 2 , 109 = 10 + 3 je takov posloupnost.) 2 Dkaz Greenovy a Taovy vty o prvoslech Pirozen sla f1; 2; : : :g ozname N a mnoinu f1; 2; : : : ; Ng, pro N 2 N, jako [N ]. Symboly Z, Q, R a C oznauj mnoiny celch, racionlnch, relnch a komplex
Mathematisches Forschungsinstitut Oberwolfach Report No. 46/2004 Theory of the Riemann Zeta and Allied Functions
, 2004
"... Introduction by the Organisers This meeting, the second Oberwolfach workshop devoted to zeta functions, was attended by 42 participants representing 16 countries. The scientific program consisted of 32 talks of various lengths and a problem session. In addition, social activities were organised: a h ..."
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Introduction by the Organisers This meeting, the second Oberwolfach workshop devoted to zeta functions, was attended by 42 participants representing 16 countries. The scientific program consisted of 32 talks of various lengths and a problem session. In addition, social activities were organised: a hike in the mountains and piano recitals by Peter
PRIMES IN TUPLES III: On the difference pn+ν − pn
"... As an approximation to the twin prime problem, Hardy and Littlewood initiated the investigation of (1.1) ∆ν: = lim inf n→∞ ..."
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As an approximation to the twin prime problem, Hardy and Littlewood initiated the investigation of (1.1) ∆ν: = lim inf n→∞
PRIME NUMBERS IN LOGARITHMIC INTERVALS
"... Abstract. Let X be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type (p, p + h], where p ≤ X is a prime number and h = o(X). Then we will apply this to prove that for every λ>1/2 there exists a positive proportion of primes p ≤ X s ..."
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Abstract. Let X be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type (p, p + h], where p ≤ X is a prime number and h = o(X). Then we will apply this to prove that for every λ>1/2 there exists a positive proportion of primes p ≤ X such that the interval (p, p+λ log X] contains at least a prime number. As a consequence we improve Cheer and Goldston’s result on the size of real numbers λ> 1 with the property that there is a positive proportion of integers m ≤ X such that the interval (m, m + λ log X] contains no primes. We also prove other results concerning the moments of the gaps between consecutive primes and about the positive proportion of integers m ≤ X such that the interval (m, m + λ log X] contains at least a prime number. The last applications of these techniques are two theorems (the first one unconditional and the second one in which we assume the validity of the Riemann Hypothesis and of a form of the Montgomery pair correlation conjecture) on the positive proportion of primes p ≤ X such that the interval (p, p + λ log X] containsnoprimes. 1.