Contact｜Sitemap｜HOME｜Japanese
HOME > Table of Contents and Abstracts > Vol. 62, No. 4
Tohoku Mathematical Journal
2010
December
SECOND SERIES VOL. 62, NO. 4
Tohoku Math. J.
62 (2010), 475483

Title
SYMMETRIC CANTOR MEASURE, COINTOSSING AND SUM SETS
Author
Gavin Brown
(Received April 3, 2009, revised April 9, 2010) 
Abstract.
Construct a probability measure $\mu$ on the circle by successive removal of middle third intervals with redistributions of the existing mass at the $n$th stage being determined by probability $p_n$ applied uniformly across that level. Assume that the sequence $\{p_n\}$ is bounded away from both $0$ and $1$. Then, for sufficiently large $N$, (estimates are given) the Lebesgue measure of any algebraic sum of Borel sets $E_1,E_2,\ldots,E_N$ exceeds the product of the corresponding $\mu(E_i)^\alpha$, where $\alpha$ is determined by $N$ and $\{p_n\}$. It is possible to replace 3 by any integer $M\geq 2$ and to work with distinct measures $\mu_1,\mu_2,\ldots,\mu_N$.
This substantially generalizes work of Williamson and the author (for powers of singlecoin cointossing measures in the case $M=2$) and is motivated by the extension to $M=3$.
We give also a simple proof of a result of Yin and the author for random variables whose binary digits are determined by cointossing.
2000 Mathematics Subject Classification.
Primary 28A60.


To the top of this page
Back to the Contents