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

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  • 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.


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  • [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.


    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.


    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.


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


    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.


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


    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.


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


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


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


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


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


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


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


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


    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.


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


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


    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.


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


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


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


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

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