Equidistribution in the dual group of the S -adic integers

Roman Urban

Czechoslovak Mathematical Journal (2014)

  • Volume: 64, Issue: 4, page 911-931
  • ISSN: 0011-4642

Abstract

top
Let X be the quotient group of the S -adele ring of an algebraic number field by the discrete group of S -integers. Given a probability measure μ on X d and an endomorphism T of X d , we consider the relation between uniform distribution of the sequence T n 𝐱 for μ -almost all 𝐱 ∈ X d and the behavior of μ relative to the translations by some rational subgroups of X d . The main result of this note is an extension of the corresponding result for the d -dimensional torus 𝕋 d due to B. Host.

How to cite

top

Urban, Roman. "Equidistribution in the dual group of the $S$-adic integers." Czechoslovak Mathematical Journal 64.4 (2014): 911-931. <http://eudml.org/doc/269860>.

@article{Urban2014,
abstract = {Let $X$ be the quotient group of the $S$-adele ring of an algebraic number field by the discrete group of $S$-integers. Given a probability measure $\mu $ on $X^d$ and an endomorphism $T$ of $X^d$, we consider the relation between uniform distribution of the sequence $T^n\mathbf \{x\}$ for $\mu $-almost all $\mathbf \{x\}\in X^d$ and the behavior of $\mu $ relative to the translations by some rational subgroups of $X^d$. The main result of this note is an extension of the corresponding result for the $d$-dimensional torus $\mathbb \{T\}^d$ due to B. Host.},
author = {Urban, Roman},
journal = {Czechoslovak Mathematical Journal},
keywords = {uniform distribution modulo $1$; equidistribution in probability; algebraic number fields; $S$-adele ring; $S$-integer dynamical system; algebraic dynamics; topological dynamics; $a$-adic solenoid; uniform distribution modulo 1; equidistribution in probability; algebraic number fields; $S$-adele ring; $S$-integer dynamical system; algebraic dynamics; topological dynamics; -adic solenoid},
language = {eng},
number = {4},
pages = {911-931},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Equidistribution in the dual group of the $S$-adic integers},
url = {http://eudml.org/doc/269860},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Urban, Roman
TI - Equidistribution in the dual group of the $S$-adic integers
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 4
SP - 911
EP - 931
AB - Let $X$ be the quotient group of the $S$-adele ring of an algebraic number field by the discrete group of $S$-integers. Given a probability measure $\mu $ on $X^d$ and an endomorphism $T$ of $X^d$, we consider the relation between uniform distribution of the sequence $T^n\mathbf {x}$ for $\mu $-almost all $\mathbf {x}\in X^d$ and the behavior of $\mu $ relative to the translations by some rational subgroups of $X^d$. The main result of this note is an extension of the corresponding result for the $d$-dimensional torus $\mathbb {T}^d$ due to B. Host.
LA - eng
KW - uniform distribution modulo $1$; equidistribution in probability; algebraic number fields; $S$-adele ring; $S$-integer dynamical system; algebraic dynamics; topological dynamics; $a$-adic solenoid; uniform distribution modulo 1; equidistribution in probability; algebraic number fields; $S$-adele ring; $S$-integer dynamical system; algebraic dynamics; topological dynamics; -adic solenoid
UR - http://eudml.org/doc/269860
ER -

References

top
  1. Berend, D., 10.1090/S0002-9947-1984-0760973-X, Trans. Am. Math. Soc. 286 (1984), 505-535. (1984) Zbl0523.22004MR0760973DOI10.1090/S0002-9947-1984-0760973-X
  2. Bertin, M.-J., Decomps-Guilloux, A., Grandet-Hugot, M., Pathiaux-Delefosse, M., Schreiber, J.-P., Pisot and Salem Numbers, With a preface by David W. Boyd Birkhäuser Basel (1992). (1992) Zbl0772.11041MR1187044
  3. Chothi, V., Everest, G., Ward, T., S -integer dynamical systems: periodic points, J. Reine Angew. Math. 489 (1997), 99-132. (1997) Zbl0879.58037MR1461206
  4. Drmota, M., Tichy, R. F., 10.1007/BFb0093405, Lecture Notes in Mathematics 1651 Springer, Berlin (1997). (1997) Zbl0877.11043MR1470456DOI10.1007/BFb0093405
  5. Gouvêa, F. Q., p -adic Numbers: An Introduction, Universitext Springer, Berlin (1997). (1997) Zbl0874.11002MR1488696
  6. Halmos, P. R., 10.1090/S0002-9904-1943-07995-5, Bull. Am. Math. Soc. 49 (1943), 619-624. (1943) Zbl0061.04403MR0008647DOI10.1090/S0002-9904-1943-07995-5
  7. Hewitt, E., Ross, K. A., Abstract Harmonic Analysis. Vol. I: Structure of topological groups, integration theory, group representations, Fundamental Principles of Mathematical Sciences 115 Springer, Berlin (1979). (1979) MR0551496
  8. Host, B., Some results of uniform distribution in the multidimensional torus, Ergodic Theory Dyn. Syst. 20 (2000), 439-452. (2000) Zbl1047.37003MR1756978
  9. Host, B., 10.1007/BF02761660, Isr. J. Math. 91 French (1995), 419-428. (1995) Zbl0839.11030MR1348326DOI10.1007/BF02761660
  10. Koblitz, N., 10.1007/978-1-4684-0047-2, Graduate Texts in Mathematics 58 Springer, New York (1977). (1977) Zbl0364.12015MR0754003DOI10.1007/978-1-4684-0047-2
  11. Kuipers, L., Niederreiter, H., Uniform Distribution of Sequences. Pure and Applied Mathematics, John Wiley & Sons New York (1974). (1974) MR0419394
  12. Mahler, K., p -adic Numbers and Their Functions, Cambridge Tracts in Mathematics 76 Cambridge University Press, Cambridge (1981). (1981) Zbl0444.12013MR0644483
  13. Narkiewicz, W., Elementary and Analytic Theory of Algebraic Numbers, Springer Monographs in Mathematics Springer, Berlin (2004). (2004) Zbl1159.11039MR2078267
  14. Neukirch, J., Algebraic Number Theory, Fundamental Principles of Mathematical Sciences 322 Springer, Berlin (1999). (1999) Zbl0956.11021MR1697859
  15. Ramakrishnan, D., Valenza, R. J., 10.1007/978-1-4757-3085-2, Graduate Texts in Mathematics 186 Springer, New York (1999). (1999) Zbl0916.11058MR1680912DOI10.1007/978-1-4757-3085-2
  16. Robert, A. M., 10.1007/978-1-4757-3254-2, Graduate Texts in Mathematics 198 Springer, New York (2000). (2000) Zbl0947.11035MR1760253DOI10.1007/978-1-4757-3254-2
  17. Schmidt, K., Dynamical Systems of Algebraic Origin, Progress in Mathematics 128 Birkhäuser, Basel (1995). (1995) Zbl0833.28001MR1345152
  18. Urban, R., Equidistribution in the d -dimensional a -adic solenoids, Unif. Distrib. Theory 6 (2011), 21-31. (2011) MR2817758
  19. Weil, A., Basic Number Theory, Die Grundlehren der Mathematischen Wissenschaften 144 Springer, New York (1974). (1974) Zbl0326.12001MR0427267

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.