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

Acta Arithmetica (1998)

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

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topY. 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

top- [B] R. C. Baker, Riemann sums and Lebesgue integrals, Quart. J. Math. Oxford Ser. (2) 27 (1976), 191-198. Zbl0333.10033
- [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.
- [Bo] J. Bourgain, Almost sure convergence and bounded entropy, Israel J. Math. 63 (1988), 79-97. Zbl0677.60042
- [BW] Y. Bugeaud and M. Weber, Examples and counterexamples for Riemann sums, preprint, I.R.M.A., Strasbourg, 1996.
- [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
- [J] B. Jessen, On the approximation of Lebesgue integrals by Riemann sums, Ann. of Math. 35 (1934), 248-251. Zbl0009.30603
- [K] U. Krengel, Ergodic Theorems, de Gruyter Stud. Math. 6, de Gruyter, Berlin, 1985.
- [M] J. M. Marstrand, On Khinchin's conjecture about strong uniform distribution, Proc. London Math. Soc. (3) 21 (1970), 540-556. Zbl0208.31402
- [N] R. Nair, On strong uniform distribution, Acta Arith. 56 (1990), 183-193. Zbl0716.11036
- [N1] R. Nair, On Riemann sums and Lebesgue integrals, Monatsh. Math. 120 (1995), 49-54. Zbl0833.28002
- [R] W. Rudin, An arithmetic property of Riemann sums, Proc. Amer. Math. Soc. 15 (1964), 321-324. Zbl0132.03601
- [T] G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, Vol. 1, Société Mathématique de France, 1995.

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