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

Abstract

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.