Let \(d\ge 2\) and \(\mathbb{S}^{d}_R\) be the \(d\)-sphere with radius \(R>0\). A measurable set \(S\subset \mathbb{S}^{d-1}\) is called \((\gamma,r)\)-thick for some small radius \(r>0\) and \(\gamma\in (0,1]\) if
\[ |S \cap K|\ge \gamma |K| \]for all spherical caps \(K\) of radius \(r\).
Note
- I could specify an upper bound for \(r\).