A1 Journal article (refereed)
On several notions of complexity of polynomial progressions (2023)
Kuca, B. (2023). On several notions of complexity of polynomial progressions. Ergodic Theory and Dynamical Systems, 43(4), 1269-1323. https://doi.org/10.1017/etds.2021.171
JYU authors or editors
Publication details
All authors or editors: Kuca, Borys
Journal or series: Ergodic Theory and Dynamical Systems
ISSN: 0143-3857
eISSN: 1469-4417
Publication year: 2023
Publication date: 20/01/2022
Volume: 43
Issue number: 4
Pages range: 1269-1323
Publisher: Cambridge University Press
Publication country: United Kingdom
Publication language: English
DOI: https://doi.org/10.1017/etds.2021.171
Publication open access: Openly available
Publication channel open access: Partially open access channel
Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/80060
Publication is parallel published: https://arxiv.org/abs/2104.07339
Abstract
For a polynomial progression (x, x + P1(y), . . . , x + Pt(y)), we define four notions of complexity: Host Kra complexity, Weyl complexity, true complexity and algebraic complexity. The first two describe the smallest characteristic factor of the progression, the third refers to the smallest-degree Gowers norm controlling the progression, and the fourth concerns algebraic relations between terms of the progressions. We conjecture that these four notions are equivalent, which would give a purely algebraic criterion for determining the smallest Host Kra factor or the smallest Gowers norm controlling a given progression. We prove this conjecture for all progressions whose terms only satisfy homogeneous algebraic relations and linear combinations thereof. This family of polynomial progressions includes, but is not limited to, arithmetic progressions, progressions with linearly independent polynomials P1, . . . ,Pt and progressions whose terms satisfy no quadratic relations. For progressions that satisfy only linear relations, such as (x, x + y2, x + 2y2, x + y3, x + 2y3), we derive several combinatorial and dynamical corollaries: first, an estimate for the count of such progressions in subsets of Z/NZ or totally ergodic dynamical systems; second, a lower bound for multiple recurrence; and third, a popular common difference result in Z/NZ. Lastly, we show that Weyl complexity and algebraic complexity always agree, which gives a straightforward algebraic description of Weyl complexity.
Keywords: number theory; combinatorics; number sequences; polynomials; dynamical systems
Free keywords: Gowers norms; Host-Kra factors; multiple recurrence; polynomial progressions
Contributing organizations
Ministry reporting: Yes
VIRTA submission year: 2022
JUFO rating: 2