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The primes contain arbitrarily long polynomial progressions
 Acta Math
"... Abstract. We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integervalued polynomials P1,..., Pk ∈ Z[m] in one unknown m with P1(0) =... = Pk(0) = 0 and any ε> 0, we show that there are infinitely many integers x, m with 1 ≤ m ≤ x ε suc ..."
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Cited by 30 (4 self)
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Abstract. We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integervalued polynomials P1,..., Pk ∈ Z[m] in one unknown m with P1(0) =... = Pk(0) = 0 and any ε> 0, we show that there are infinitely many integers x, m with 1 ≤ m ≤ x ε such that x+P1(m),..., x+Pk(m) are simultaneously prime. The arguments are based on those in [18], which treated the linear case Pi = (i − 1)m and ε = 1; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number of points over finite fields in certain algebraic varieties. Contents
The GreenTao Theorem on arithmetic progressions in the primes: an ergodic point of view
, 2005
"... A longstanding and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an a ..."
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Cited by 18 (2 self)
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A longstanding and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions. Until recently, the only progress on this conjecture was due to van der Corput, who showed in 1939 that there are infinitely many triples of primes in arithmetic progression. In an amazing fusion of methods from analytic number theory and ergodic theory, Ben Green and Terence Tao showed that for any positive integer k, there exist infinitely many arithmetic progressions of length k consisting only of prime numbers. This is an introduction to some of the ideas in the proof, concentrating on the connections to ergodic theory.
Multiple ergodic averages for three polynomials and applications
, 2006
"... Abstract. We find the smallest characteristic factor and a limit formula for the multiple ergodic averages associated to any family of three polynomials and polynomial families of the form {p, 2p,..., kp}. We then derive several combinatorial implications, including an answer to a question of Brown, ..."
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Cited by 9 (2 self)
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Abstract. We find the smallest characteristic factor and a limit formula for the multiple ergodic averages associated to any family of three polynomials and polynomial families of the form {p, 2p,..., kp}. We then derive several combinatorial implications, including an answer to a question of Brown, Graham, and Landman, and a generalization of the Polynomial Szemerédi Theorem of Bergelson and Leibman for families of three polynomials with not necessarily zero constant term. We also simplify and generalize a recent result of Bergelson, Host, and Kra, showing that for all ε> 0 and every subset of the integers Λ the set n ∈ N: d ∗ ( Λ ∩ (Λ + p1(n)) ∩ (Λ + p2(n)) ∩ (Λ + p3(n)) )> (d ∗ (Λ)) 4 − ε} has bounded gaps for “most ” choices of integer polynomials p1, p2, p3. Contents
Intersective polynomials and polynomial Szemerédi theorem
, 2008
"... Let P = {p1,..., pr} ⊂ Q[n1,..., nm] be a family of polynomials such that pi(Zm) ⊆ Z, i = 1,..., r. We say that the family P has the PSZ property if for any set E ⊆ Z with d ∗ E∩[M,N−1] (E) = lim supN−M→ ∞ N−M> 0 there exist infinitely many n ∈ Zm such that E contains a polynomial progression o ..."
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Cited by 7 (1 self)
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Let P = {p1,..., pr} ⊂ Q[n1,..., nm] be a family of polynomials such that pi(Zm) ⊆ Z, i = 1,..., r. We say that the family P has the PSZ property if for any set E ⊆ Z with d ∗ E∩[M,N−1] (E) = lim supN−M→ ∞ N−M> 0 there exist infinitely many n ∈ Zm such that E contains a polynomial progression of the form {a, a + p1(n),..., a + pr(n)}. We prove that a polynomial family P = {p1,..., pr} has the PSZ property if and only if the polynomials p1,..., pr are jointly intersective, meaning that for any k ∈ N there exists n ∈ Zm such that the integers p1(n),..., pr(n) are all divisible by k. To obtain this result we give a new ergodic proof of the polynomial Szemerédi theorem, based on the fact that the key to the phenomenon of polynomial multiple recurrence lies with the dynamical systems defined by translations on nilmanifolds. We also obtain, as a corollary, the following generalization of the polynomial van der Waerden theorem: If p1,..., pr ∈ Q[n] are jointly intersective integral polynomials, then for any finite partition Z = ⋃k i=1 Ei of Z, there exist i ∈ {1,..., k} and a, n ∈ Ei such that {a, a+p1(n),..., a+pr(n)} ⊂ Ei.
Powers of Sequences and Recurrence
, 2008
"... We study recurrence, and multiple recurrence, properties along the kth powers of a given set of integers. We show that the property of recurrence for some given values of k does not give any constraint on the recurrence for the other powers. This is motivated by similar results in number theory co ..."
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Cited by 5 (5 self)
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We study recurrence, and multiple recurrence, properties along the kth powers of a given set of integers. We show that the property of recurrence for some given values of k does not give any constraint on the recurrence for the other powers. This is motivated by similar results in number theory concerning additive basis of natural numbers. Moreover, motivated by a result of Kamae and MendèsFrance, that links single recurrence with uniform distribution properties of sequences, we look for an analogous result dealing with higher order recurrence and make a related conjecture.
MULTIPLE RECURRENCE AND CONVERGENCE FOR SEQUENCES RELATED TO THE PRIME NUMBERS
"... Abstract. For any measure preserving system (X, X, µ, T) and A ∈ X with µ(A)> 0, we show that there exist infinitely many primes p such that µ ` A ∩ T −(p−1) A ∩ T −2(p−1) A ´> 0 (the same holds with p − 1 replaced by p + 1). Furthermore, we show the existence of the limit in L 2 (µ) of the associat ..."
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Cited by 5 (1 self)
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Abstract. For any measure preserving system (X, X, µ, T) and A ∈ X with µ(A)> 0, we show that there exist infinitely many primes p such that µ ` A ∩ T −(p−1) A ∩ T −2(p−1) A ´> 0 (the same holds with p − 1 replaced by p + 1). Furthermore, we show the existence of the limit in L 2 (µ) of the associated ergodic average over the primes. A key ingredient is a recent result of Green and Tao on the von Mangoldt function. A combinatorial consequence is that every subset of the integers with positive upper density contains an arithmetic progression of length three and common difference of the form p − 1 (or p + 1) for some prime p. 1.
ERGODIC AVERAGES FOR INDEPENDENT POLYNOMIALS AND APPLICATIONS
, 2006
"... Abstract. Szemerédi’s Theorem states that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman generalized this, showing that sets of integers with positive upper density contain arbitrarily long polynomial configurations; Szemerédi’s ..."
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Cited by 5 (3 self)
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Abstract. Szemerédi’s Theorem states that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman generalized this, showing that sets of integers with positive upper density contain arbitrarily long polynomial configurations; Szemerédi’s Theorem corresponds to the linear case of the polynomial theorem. We focus on the case farthest from the linear case, that of rationally independent polynomials. We derive results in ergodic theory and in combinatorics for rationally independent polynomials, showing that their behavior differs sharply from the general situation.
A HARDY FIELD EXTENSION OF SZEMERÉDI’S THEOREM
, 2008
"... In 1975 Szemerédi proved that a set of integers of positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman showed in 1996 that the common difference of the arithmetic progression can be a square, a cube, or more generally of the form p(n) where p(n) is any i ..."
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Cited by 4 (1 self)
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In 1975 Szemerédi proved that a set of integers of positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman showed in 1996 that the common difference of the arithmetic progression can be a square, a cube, or more generally of the form p(n) where p(n) is any integer polynomial with zero constant term. We produce a variety of new results of this type related to sequences that are not polynomial. We show that the common difference can be of the form [n δ] where δ is any positive real number and [x] denotes the integer part of x. More generally, the common difference can be of the form [a(n)] where a(x) is any function from a Hardy field which is sandwiched between two consecutive powers of x, that is, a(x)/x k → ∞ and a(x)/x k+1 → 0 for some nonnegative integer k. This allows us for example to deal with functions that can be constructed by a finite combination of the ordinary arithmetical symbols, the real constants, the real variable x, and the functional symbols exp and log, and satisfy the previous growth assumptions. The proof combines a new structural result for Hardy sequences, techniques from ergodic theory, and some recent equidistribution results of sequences on nilmanifolds.