Multiple recurrence and convergence for sequences related to the prime numbers

By Nikos Frantzikinakis, B. Host, Bryna Kra. 2006

65 citations recorded by the source corpus.

For any measure preserving system (X, , μ,T) and A ∈ with μ(A) > 0, we show that there exist infinitely many primes p such that (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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/66a23b3ebf0eda251197f1b386a41d64a49e381e

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Citations in the corpus

THE LOGARITHMICALLY AVERAGED CHOWLA AND ELLIOTT CONJECTURES FOR TWO-POINT CORRELATIONS (2015, methodology): “Using the literature from higher order Fourier analysis (in particular the inverse theorem in [15], together with transference arguments from [9, 14], or [33]), one is now faced with the task of controlling sums even more complicated than (51), in which the linear phases n 7→ e(αn) are now replaced ”
Ergodic seminorms for commuting transformations and applications (2008, methodology): “The same structure can be used to study other problems of multiple convergence and of multiple recurrence, for example in [HK2], [L], [BHK], [ FHK ], [FK2], ... A similar method was used by Conze & Lesigne for two commuting transformations, but all attempts to solve the general case by using the mac”
An inverse theorem for the Gowers U^{s+1}[N]-norm (announcement) (2010, background): “Several further applications of the GI(s) conjectures are given in [6, 18].”
Equivalence of the Logarithmically Averaged Chowla and Sarnak Conjectures (2016, methodology): “Manipulations of this form have appeared in [22], [59]; we will use an argument somewhat similar to that in [22]:”
The logarithmic Sarnak conjecture for ergodic weights (2017, result): “This result is not stated explicitly in [23], but follows from the argument in [23, §5], using Theorem 4.”