On the Ehrenfeucht-Mycielski Balance Conjecture
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07), DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07) (2007)
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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.
@article{DMTCS_2007_special_253_a24,
author = {Kieffer, John C. and Szpankowski, W.},
title = {On the {Ehrenfeucht-Mycielski} {Balance} {Conjecture}},
journal = {Discrete mathematics & theoretical computer science},
year = {2007},
volume = {DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07)},
doi = {10.46298/dmtcs.3542},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3542/}
}
TY - JOUR AU - Kieffer, John C. AU - Szpankowski, W. TI - On the Ehrenfeucht-Mycielski Balance Conjecture JO - Discrete mathematics & theoretical computer science PY - 2007 VL - DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07) UR - http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3542/ DO - 10.46298/dmtcs.3542 LA - en ID - DMTCS_2007_special_253_a24 ER -
%0 Journal Article %A Kieffer, John C. %A Szpankowski, W. %T On the Ehrenfeucht-Mycielski Balance Conjecture %J Discrete mathematics & theoretical computer science %D 2007 %V DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07) %U http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3542/ %R 10.46298/dmtcs.3542 %G en %F DMTCS_2007_special_253_a24
Kieffer, John C.; Szpankowski, W. On the Ehrenfeucht-Mycielski Balance Conjecture. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07), DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07) (2007). doi: 10.46298/dmtcs.3542
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