Gold has a molar mass of 196.97 g mol⁻¹…

Step 1. One atom. $\dfrac{196.97\ \text{g}}{6.022 \times 10^{23}} = 3.271 \times 10^{-22}\ \text{g}$.

Step 2. Moles in the ring. $\dfrac{3.94}{196.97} = 0.02000\ \text{mol}$.

Step 3. Atoms in the ring. $0.02000 \times 6.022 \times 10^{23} = 1.205 \times 10^{22}$ atoms.