A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
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-tekijät tai -toimittajat

Julkaisun tiedot

Julkaisun kaikki tekijät tai toimittajat: Kuca, Borys

Lehti tai sarja: Ergodic Theory and Dynamical Systems

ISSN: 0143-3857

eISSN: 1469-4417

Julkaisuvuosi: 2023

Ilmestymispäivä: 20.01.2022

Volyymi: 43

Lehden numero: 4

Artikkelin sivunumerot: 1269-1323

Kustantaja: Cambridge University Press

Julkaisumaa: Britannia

Julkaisun kieli: englanti

DOI: https://doi.org/10.1017/etds.2021.171

Julkaisun avoin saatavuus: Avoimesti saatavilla

Julkaisukanavan avoin saatavuus: Osittain avoin julkaisukanava

Julkaisu on rinnakkaistallennettu (JYX): https://jyx.jyu.fi/handle/123456789/80060

Julkaisu on rinnakkaistallennettu: https://arxiv.org/abs/2104.07339

Tiivistelmä

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.

YSO-asiasanat: lukuteoria; kombinatoriikka; lukujonot; polynomit; dynaamiset systeemit

Vapaat asiasanat: Gowers norms; Host-Kra factors; multiple recurrence; polynomial progressions

Liittyvät organisaatiot

OKM-raportointi: Kyllä

VIRTA-lähetysvuosi: 2022

JUFO-taso: 2