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July  2013, 7(3): 461-488. doi: 10.3934/jmd.2013.7.461

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

Francesco Cellarosi 1, and  Ilya Vinogradov 2,

1. 

Department of Mathematics, Altgeld Hall, 1409 W Green Street, Urbana, IL 61801, United States
2. 

School of Mathematics, University Walk, Bristol, BS8 1TW, United Kingdom

Received  March 2013 Revised  September 2013 Published  December 2013

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 $k=2$.
Keywords: ergodicity, group actions with pure point spectrum, isomorphism of group actions., Square-free and $k$-free integers in number fields, correlation functions.
Mathematics Subject Classification: Primary: 37A35, 37A45; Secondary: 11R04, 11N25, 37C85, 28D1.
Citation: Francesco Cellarosi, Ilya Vinogradov. Ergodic properties of $k$-free integers in number fields. Journal of Modern Dynamics, 2013, 7 (3) : 461-488. doi: 10.3934/jmd.2013.7.461
References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[9]

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

[10]

J. Liu and P. Sarnak, The Möbius function and distal flows,, preprint., ().   Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

P. Sarnak, Three lectures on the Möbius function randomness and dynamics (Lecture 1)., Available at: , ().   Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[9]

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

[10]

J. Liu and P. Sarnak, The Möbius function and distal flows,, preprint., ().   Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[19]

P. Sarnak, Three lectures on the Möbius function randomness and dynamics (Lecture 1)., Available at: , ().   Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

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