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LEBESGUE MEASURE OF SUM SETS – THE BASIC RESULT FOR COIN-TOSSING

Published online by Cambridge University Press:  19 May 2004

GAVIN BROWN
Affiliation:
Vice-Chancellor, The University of Sydney, NSW 2006, Australia e-mail: Vice-Chancellor@usyd.edu.au
QINGHE YIN
Affiliation:
Institute for Infocomm Research, 21 Heng Mui Keng Terrace, Singapore 119613 e-mail: yinqh@i2r.a-star.edu.sg
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Abstract

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Let $\mu_p$ be the distribution of a random variable on the interval $[0,1)$, each digit of whose binary expansion is 0 or 1 with probability $p$ or $1\,{-}\,p$. Thus $\mu_p\,{=}\,\mathop{*}^{\infty}_{n{=}1} (p\delta_0+(1-p)\delta_{\frac1{2^n}})$. We show that for any Borel subsets $E$, $F$ of $[0,1)$ we have $$\l(E+F)\ge\mu_p(E)^\a\mu_q(F)^\b,$$ where $0\,{<}\,\a, \b\,{<}\,1$ with $\a\log a+\b\log b\,{=}\,\log 2$ and $a\,{=}\,[\max\{p, 1-p\}]^{-1}$, $b\,{=}\,[\max\{q, 1-q\}]^{-1}$. Here $\l=\mu_{1/2}$ denotes Lebesgue measure.

Type
Research Article
Copyright
2004 Glasgow Mathematical Journal Trust