## Real Analysis Exchange

### The relative growth of information in two-dimensional partitions.

#### Abstract

Let $\overline{x} \in [0,1)^2$. In this paper we find the rate at which knowledge about the partition elements $\overline{x}$ lies in for one sequence of partitions determines the partition elements it lies in for another sequence of partitions. This rate depends on the entropy of these partitions and the geometry of their shapes, and gives a two-dimensional version of Lochs' theorem.

#### Article information

Source
Real Anal. Exchange Volume 31, Number 2 (2005), 397-408.

Dates
First available in Project Euclid: 10 July 2007

https://projecteuclid.org/euclid.rae/1184104033

Mathematical Reviews number (MathSciNet)
MR2265782

Zentralblatt MATH identifier
1107.28011

Subjects
Primary: 28D15: General groups of measure-preserving transformations
Secondary: 28D20: Entropy and other invariants

#### Citation

Dajani, Karma; de Vries, Martijn; Johnson, Aimee S. A. The relative growth of information in two-dimensional partitions. Real Anal. Exchange 31 (2005), no. 2, 397--408.https://projecteuclid.org/euclid.rae/1184104033

#### References

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• W. Bosma, K. Dajani, C. Kraaikamp, Entropy and Counting Correct Digits, University of Nijmegen, report no. 9925, 1999, http://www.math.kun.nl/onderzoek/reports/reports1999.html
• K. Dajani, A. Fieldsteel, Equipartition of Interval Partitions and an Application to Number Theory, Proc. Amer. Math. Soc., 129, 12, (2001), 3453–3460.
• E. Lindenstrauss, Pointwise Theorems for Amenable Groups, Inventiones Mathematicae, 146 (2001), 259–295.
• G. Lochs, Vergleich der Genauigkeit von Dezimalbruch und Kettenbruch, Abh. Math. Sem. Hamburg, 27 (1964), 142–144.