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Modified Schmidt games and a conjecture of Margulis
Ergodic properties of $k$-free integers in number fields
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 |
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. |
[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: , ().
|
[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. |
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. |
[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: , ().
|
[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. |
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