# On the Ehrenfeucht-Mycielski Balance Conjecture

Abstract : In 1992, A. Ehrenfeucht and J. Mycielski defined a seemingly pseudorandom binary sequence which has since been termed the EM-sequence. The balance conjecture for the EM-sequence, still open, is the conjecture that the sequence of EM-sequence initial segment averages converges to $1/2$. In this paper, we do not prove the balance conjecture but we do make some progress concerning it, namely, we prove that every limit point of the aforementioned sequence of averages lies in the interval $[1/4,3/4]$, improving the best previous result that every such limit point belongs to the interval $[0.11,0.89]$. Our approach is novel and exploits an analysis of the growth behavior as $n \to \infty$ of the rooted tree formed by the binary strings appearing at least twice as substrings of the length $n$ initial segment of the EM-sequence.
Keywords :
Type de document :
Communication dans un congrès
Jacquet, Philippe. 2007 Conference on Analysis of Algorithms, AofA 07, 2007, Juan les Pins, France. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07), pp.19-30, 2007, DMTCS Proceedings
Domaine :

Littérature citée [5 références]

https://hal.inria.fr/hal-01184790
Contributeur : Coordination Episciences Iam <>
Soumis le : lundi 17 août 2015 - 16:59:54
Dernière modification le : jeudi 11 mai 2017 - 01:02:52
Document(s) archivé(s) le : mercredi 18 novembre 2015 - 12:17:33

### Fichier

dmAH0102.pdf
Fichiers éditeurs autorisés sur une archive ouverte

### Identifiants

• HAL Id : hal-01184790, version 1

### Citation

John C. Kieffer, W. Szpankowski. On the Ehrenfeucht-Mycielski Balance Conjecture. Jacquet, Philippe. 2007 Conference on Analysis of Algorithms, AofA 07, 2007, Juan les Pins, France. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07), pp.19-30, 2007, DMTCS Proceedings. 〈hal-01184790〉

### Métriques

Consultations de la notice

## 62

Téléchargements de fichiers