The distance set of a subset E of \(R^ n\) is \(D(E)=\{| x- y|:x,y\in E\}.\) If E is analytic (i.e. Suslin), the author uses Fourier transform to derive the following lower bound for the Hausdorff dimension of \(E\): \[ \dim D(E)\geq \min \{1,(\dim E)-(n-1)/2\}. \] Moreover, \(D(E)\) has positive Lebesgue measure if \(\dim E>(n+1)/2\). The continuum hypothesis is used to show that for general non-analytic sets no such results hold.