Nilsequences and multiple correlations along subsequences

Abstract

The results of Bergelson-Host-Kra and Leibman say that a multiple polynomial correlation sequence can be decomposed into a sum of a nilsequence (a sequence defined by evaluating a continuous function along an orbit in a nilsystem) and a null sequence (a sequence that goes to zero in density). We refine their results by proving that the null sequence goes to zero in density along polynomials evaluated at primes and Hardy sequence (⌊n c ⌋). On the other hand, given a rigid sequence, we construct an example of correlation whose null sequence does not approach zero in density along that rigid sequence. As a corollary of a lemma in the proof, the formula for the pointwise ergodic average along polynomials of primes in a nilsystem is also obtained.

Interpolation sets and nilsequences

“…Hence there is a question related to Question 1.1: Is it true that lim N →∞ 1 N N n=1 |a(r n )−ψ a (r n )| = 0 for r n = p n , n c or 2 n ? This question is answered affirmatively for (p n ) n∈N and ( n c ) n∈N in [20]. However, it is negative for (2 n ) n∈N , which is in contrast with the answer for Question 1.1.…”

“…Inspired by these results, Frantzikinakis asks the following question: Question 13] Let r n = p n (n-th prime), n c for some c > 0, or 2 n . Is it true that for any k-multiple correlation (a(n)) n∈N , there exists a uniform limit of k-step nilsequence (ψ(n)) n∈N such that In [20], we give affirmative answers for r n = p n and n c (The case of the primes p n is also proved by Tao and Teräväinen [33]). Positive answers for these two sequences are expected because they share important properties with the full sequence of natural numbers r n = n. More specifically, we can use the Host-Kra Structure Theorem [17,34] to project the averages of multiple correlations along these two sequences to the nilfactors without affecting the averages.…”

“…The answer is probably negative. [20] that the answer to Question 1.1 is affirmative for all r n = P (n) or P (p n ) where P ∈ Z[n] non-constant, and for (r n ) n∈N in a large class of Hardy field sequences, for example, r n = n log n or n 2 √ 2 + n √ 3 . By…”

Odd order cases of the logarithmically averaged Chowla conjecture

“…If we set c p ∈ {−1, 0, +1} to be the signum of E log n≤x λ(a 1 n + apb 1 ) · · · λ(a k n + apb k )(p1 p|n − 1), it will thus suffice to show that (17) E 2 m <p≤2 m+1 c p E log x ε <n≤x λ(a 1 n + apb 1 ) · · · λ(a k n + apb k )(p1 p|n − 1) = O(ε) for all m obeying (16), outside of an exceptional set M obeying (6). Let m obey (16). If j is a natural number less than or equal to 2 m (and hence of size O(log 1/10 x)), one easily computes the total variation bound…”

“…The proof in [23] of the odd order cases of the logarithmically averaged Chowla conjecture relies on deep results of Leibman [17] and Le [16] on ergodic theory, and is not much simpler than the proof of the structural theorem for correlations of general bounded multiplicative functions in that paper. Here we give a different, shorter proof of the odd order cases of Chowla's conjecture, which avoids all use of ergodic theory, although it now requires the Gowers uniformity of the von Mangoldt function, established by Green, the first author and Ziegler [10], [11], [12].…”