On strong uniform distribution, II. The infinite-dimensional case

Y. Lacroix

Acta Arithmetica (1998)

  • Volume: 84, Issue: 3, page 279-290
  • ISSN: 0065-1036

Abstract

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We construct infinite-dimensional chains that are L¹ good for almost sure convergence, which settles a question raised in this journal [N]. We give some conditions for a coprime generated chain to be bad for L² or L ∞ , using the entropy method. It follows that such a chain with positive lower density is bad for L ∞ . There also exist such bad chains with zero density.

How to cite

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Y. Lacroix. "On strong uniform distribution, II. The infinite-dimensional case." Acta Arithmetica 84.3 (1998): 279-290. <http://eudml.org/doc/207146>.

@article{Y1998,
abstract = {We construct infinite-dimensional chains that are L¹ good for almost sure convergence, which settles a question raised in this journal [N]. We give some conditions for a coprime generated chain to be bad for L² or $L^∞$, using the entropy method. It follows that such a chain with positive lower density is bad for $L^∞$. There also exist such bad chains with zero density.},
author = {Y. Lacroix},
journal = {Acta Arithmetica},
keywords = {dimension; chains; almost sure convergence; universally good; density; infinite-dimensional chains; coprime generated chain; entropy method; bad chains},
language = {eng},
number = {3},
pages = {279-290},
title = {On strong uniform distribution, II. The infinite-dimensional case},
url = {http://eudml.org/doc/207146},
volume = {84},
year = {1998},
}

TY - JOUR
AU - Y. Lacroix
TI - On strong uniform distribution, II. The infinite-dimensional case
JO - Acta Arithmetica
PY - 1998
VL - 84
IS - 3
SP - 279
EP - 290
AB - We construct infinite-dimensional chains that are L¹ good for almost sure convergence, which settles a question raised in this journal [N]. We give some conditions for a coprime generated chain to be bad for L² or $L^∞$, using the entropy method. It follows that such a chain with positive lower density is bad for $L^∞$. There also exist such bad chains with zero density.
LA - eng
KW - dimension; chains; almost sure convergence; universally good; density; infinite-dimensional chains; coprime generated chain; entropy method; bad chains
UR - http://eudml.org/doc/207146
ER -

References

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  1. [B] R. C. Baker, Riemann sums and Lebesgue integrals, Quart. J. Math. Oxford Ser. (2) 27 (1976), 191-198. Zbl0333.10033
  2. [Be] T. Bewley, Extension of the Birkhoff and von Neumann ergodic theorems to semigroup actions, Ann. Inst. H. Poincaré Sect. B 7 (1971), 248-260. 
  3. [Bo] J. Bourgain, Almost sure convergence and bounded entropy, Israel J. Math. 63 (1988), 79-97. Zbl0677.60042
  4. [BW] Y. Bugeaud and M. Weber, Examples and counterexamples for Riemann sums, preprint, I.R.M.A., Strasbourg, 1996. 
  5. [DP] L. E. Dubins and J. Pitman, A pointwise ergodic theorem for the group of rational rotations, Trans. Amer. Math. Soc. 251 (1979), 299-308. Zbl0412.60050
  6. [J] B. Jessen, On the approximation of Lebesgue integrals by Riemann sums, Ann. of Math. 35 (1934), 248-251. Zbl0009.30603
  7. [K] U. Krengel, Ergodic Theorems, de Gruyter Stud. Math. 6, de Gruyter, Berlin, 1985. 
  8. [M] J. M. Marstrand, On Khinchin's conjecture about strong uniform distribution, Proc. London Math. Soc. (3) 21 (1970), 540-556. Zbl0208.31402
  9. [N] R. Nair, On strong uniform distribution, Acta Arith. 56 (1990), 183-193. Zbl0716.11036
  10. [N1] R. Nair, On Riemann sums and Lebesgue integrals, Monatsh. Math. 120 (1995), 49-54. Zbl0833.28002
  11. [R] W. Rudin, An arithmetic property of Riemann sums, Proc. Amer. Math. Soc. 15 (1964), 321-324. Zbl0132.03601
  12. [T] G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, Vol. 1, Société Mathématique de France, 1995. 

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