Multiple recurrence and convergence for sequences related to the prime numbers

Abstract

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.

The Logarithmically Averaged Chowla and Elliott Conjectures for Two-Point Correlations

“…Some condition of the form (8) must be needed in order to derive the conclusion (9), as one can see by considering examples such as g 1 (n) := χ (n)n it and g 2 (n) := g 1 (n), where χ is a Dirichlet character of bounded conductor, t is a real number of size t = o(x), and w is set equal to (for instance) (x/|t|) 1/2 . More precise asymptotics of sums such as those in (9) in the 'pretentious' case when g 1 and g 2 both behave like twisted Dirichlet characters n → χ (n)n it were computed in the recent preprint of Klurman [20].…”

From harmonic analysis to arithmetic combinatorics

“…The Green-Tao theorem has inspired a variety of other results in ergodic theory and number theory, such as Frantzikinakis-Host-Kra [61] or Tao-Ziegler [188]. The focus of this note will remain on connections to harmonic analysis, and thus we return to restriction theory for the last time.…”

Parallelepipeds, nilpotent groups and Gowers norms

“…The edge group G [3,1] is the subgroup of G [3] spanned by the elements of the form g [3,e] where g ∈ G and e is an edge of the cube {0, 1} 3 . The face group G [3,2] is the subgroup of G [3] spanned by the elements of the form g [3,f ] where g ∈ G and f is a face of the cube {0, 1} 3 .…”

“…-Let G be an abelian group, let P = G [2,1] be the strong parallelogram structure defined as in Example 1 and let Q = G [3,2] . Then (P, Q) is a (strong) parallelepiped structure on G.…”