No CrossRef data available.
Article contents
Multiple correlation sequences not approximable by nilsequences
Part of:
Sequences and sets
Published online by Cambridge University Press: 16 July 2021
Abstract
We show that there is a measure-preserving system $(X,\mathscr {B}, \mu , T)$ together with functions $F_0, F_1, F_2 \in L^{\infty }(\mu )$ such that the correlation sequence $C_{F_0, F_1, F_2}(n) = \int _X F_0 \cdot T^n F_1 \cdot T^{2n} F_2 \, d\mu $ is not an approximate integral combination of $2$ -step nilsequences.
Keywords
MSC classification
Secondary:
11B25: Arithmetic progressions
- Type
- Original Article
- Information
- Copyright
- © The Author(s), 2021. Published by Cambridge University Press
References
Altman, D.. On Szemerédi’s theorem with differences from a random set. Acta Arith. 195(1) (2020), 97–108.CrossRefGoogle Scholar
Bergelson, V., Host, B. and Kra, B.. Multiple recurrence and nilsequences. Invent. Math. 160(2) (2005), 261–303.CrossRefGoogle Scholar
Bergelson, V. and Leibman, A.. Distribution of values of bounded generalised polynomials. Acta Math. 198(2) (2007), 155–230.CrossRefGoogle Scholar
Briët, J. and Gopi, S.. Gaussian width bounds with applications to arithmetic progressions in random settings. Int. Math. Res. Not. 22 (2020), 8673–8696.Google Scholar
Briët, J. and Labib, F.. High-entropy dual functions over finite fields and locally decodable codes. Forum Math. Sigma 9 (2021), e19.CrossRefGoogle Scholar
Frantzikinakis, N.. Equidistribution of sparse sequences on nilmanifolds. J. Anal. Math. 109 (2009), 353–395.CrossRefGoogle Scholar
Frantzikinakis, N.. Some open problems on multiple ergodic averages. Bull. Hellenic Math. Soc. 60 (2016), 41–90.Google Scholar
Frantzikinakis, N.. An averaged Chowla and Elliott conjecture along independent polynomials. Int. Math. Res. Not. (IMRN) 12 (2018), 3721–3743.Google Scholar
Yekhanin, S.. Towards 3-query locally decodable codes of subexponential length. J. ACM 55(1) (2008), Art. 1, 16 pp.CrossRefGoogle Scholar