LEBESGUE MEASURE OF SUM SETS – THE BASIC RESULT FOR COIN-TOSSING
Glasgow mathematical journal, Tome 46 (2004) no. 2, pp. 345-353
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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.
BROWN, GAVIN; YIN, QINGHE. LEBESGUE MEASURE OF SUM SETS – THE BASIC RESULT FOR COIN-TOSSING. Glasgow mathematical journal, Tome 46 (2004) no. 2, pp. 345-353. doi: 10.1017/S001708950400182X
@article{10_1017_S001708950400182X,
author = {BROWN, GAVIN and YIN, QINGHE},
title = {LEBESGUE {MEASURE} {OF} {SUM} {SETS} {\textendash} {THE} {BASIC} {RESULT} {FOR} {COIN-TOSSING}},
journal = {Glasgow mathematical journal},
pages = {345--353},
year = {2004},
volume = {46},
number = {2},
doi = {10.1017/S001708950400182X},
url = {http://geodesic.mathdoc.fr/articles/10.1017/S001708950400182X/}
}
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