\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Ergodic properties of $k$-free integers in number fields

Abstract Related Papers Cited by
  • Let $K/\mathbf{Q}$ be a degree-$d$ extension. Inside the ring of integers $\mathscr O_K$ we define the set of $k$-free integers $\mathscr F_k$ and a natural $\mathscr O_K$-action on the space of binary $\mathscr O_K$-indexed sequences, equipped with an $\mathscr O_K$-invariant probability measure associated to $\mathscr F_k$. We prove that this action is ergodic, has pure point spectrum, and is isomorphic to a $\mathbf Z^d$-action on a compact abelian group. In particular, it is not weakly mixing and has zero measure-theoretical entropy. This work generalizes the work of Cellarosi and Sinai [J. Eur. Math. Soc. (JEMS) 15 (2013), no. 4, 1343--1374] that considered the case $K=\mathbf{Q}$ and $k=2$.
    Mathematics Subject Classification: Primary: 37A35, 37A45; Secondary: 11R04, 11N25, 37C85, 28D15.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    M. Baake, R. V. Moody and P. A. B. Pleasants, Diffraction from visible lattice points and $k$th power free integers, Discrete Math., 221 (2000), 3-42.doi: 10.1016/S0012-365X(99)00384-2.

    [2]

    V. Bergelson and A. Gorodnik, Weakly mixing group actions: A brief survey and an example, in Modern Dynamical Systems and Applications, Cambridge Univ. Press, Cambridge, 2004, 3-25.

    [3]

    F. Cellarosi and Ya. G. Sinai, Ergodic properties of square-free numbers, J. Eur. Math. Soc. (JEMS), 15 (2013), 1343-1374.doi: 10.4171/JEMS/394.

    [4]

    R. R. Hall, The distribution of squarefree numbers, J. Reine Angew. Math., 394 (1989), 107-117.doi: 10.1515/crll.1989.394.107.

    [5]

    P. R. Halmos and J. von Neumann, Operator methods in classical mechanics. II, Ann. of Math. (2), 43 (1942), 332-350.doi: 10.2307/1968872.

    [6]

    D. R. Heath-Brown, The square sieve and consecutive square-free numbers, Math. Ann., 266 (1984), 251-259.doi: 10.1007/BF01475576.

    [7]

    E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. Vol. I. Structure of Topological Groups, Integration Theory, Group Representations, Second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 115, Springer-Verlag, Berlin-New York, 1979.

    [8]

    I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Second edition, Graduate Texts in Mathematics, 113, Springer-Verlag, New York, 1991.

    [9]

    L. B. Koralov and Y. G. Sinai, Theory of probability and random processes, Second edition, Universitext, Springer, Berlin, 2007.

    [10]

    J. Liu and P. Sarnak, The Möbius function and distal flows, preprint.

    [11]

    G. W. Mackey, Ergodic transformation groups with a pure point spectrum, Illinois J. Math., 8 (1964), 593-600.

    [12]

    L. Mirsky, Arithmetical pattern problems relating to divisibility by $r$th powers, Proc. London Math. Soc. (2), 50 (1949), 497-508.

    [13]

    W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Third edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004.

    [14]

    J. Neukirch, Algebraic Number Theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 322, Springer-Verlag, Berlin, 1999.

    [15]

    R. Peckner, Uniqueness of the measure of maximal entropy for the squarefree flow, preprint, 2012.

    [16]

    P. A. B. Pleasants and C. Huck, Entropy and diffraction of the $k$-free points in $n$-dimensional lattices, Discrete Comput. Geom., 50 (2013), 39-68.doi: 10.1007/s00454-013-9516-y.

    [17]

    V. A. Rokhlin, On the problem of the classification of automorphisms of Lebesgue spaces, Doklady Akad. Nauk SSSR (N. S.), 58 (1947), 189-191.

    [18]

    V. A. Rokhlin, Unitary rings, Doklady Akad. Nauk SSSR (N. S.), 59 (1948), 643-646.

    [19]

    P. Sarnak, Three lectures on the Möbius function randomness and dynamics (Lecture 1). Available at: http://publications.ias.edu/sites/default/files/MobiusFunctionsLectures(2).pdf.

    [20]

    K. Schmidt, Dynamical Systems of Algebraic Origin, [2011 reprint of the 1995 original] [MR1345152], Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1995.

    [21]

    K. M. Tsang, The distribution of $r$-tuples of squarefree numbers, Mathematika, 32 (1985), 265-275.doi: 10.1112/S0025579300011049.

    [22]

    J. von Neumann, Zur Operatorenmethode in der klassischen Mechanik, Ann. of Math. (2), 33 (1932), 587-642.doi: 10.2307/1968537.

    [23]

    R. J. Zimmer, Ergodic actions with generalized discrete spectrum, Illinois J. Math., 20 (1976), 555-588.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(133) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return