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 editorsKuca, Borys

Journal or seriesErgodic Theory and Dynamical Systems

ISSN0143-3857

eISSN1469-4417

Publication year2023

Publication date20/01/2022

Volume43

Issue number4

Pages range1269-1323

PublisherCambridge University Press

Publication countryUnited Kingdom

Publication languageEnglish

DOIhttps://doi.org/10.1017/etds.2021.171

Publication open accessOpenly available

Publication channel open accessPartially open access channel

Publication is parallel published (JYX)https://jyx.jyu.fi/handle/123456789/80060

Publication is parallel publishedhttps://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.


Keywordsnumber theorycombinatoricsnumber sequencespolynomialsdynamical systems

Free keywordsGowers norms; Host-Kra factors; multiple recurrence; polynomial progressions


Contributing organizations


Ministry reportingYes

VIRTA submission year2022

JUFO rating2


Last updated on 2024-12-10 at 16:00