Suppose \((M,g)\) is a Riemannian manifold . Let \(\gamma>0\) and \(r>0\). A measurable set \(S \subset M\) is called \((\gamma, r)\)-thick if for every \(p\in M\)

\begin{equation*} \gamma \le \frac{\Vol(S\cap B_r(p))}{\Vol(B_r(p))} \end{equation*}

See also Link to heading

• euclidean thick set

Questions? Link to heading

• Do we need additional assumptions for \(M\)? (Perhaps, \(r\) must be smaller then the injectivity radius).