Definition ($alpha(G,eps)$). Write

$$\alpha (G,𝜀)=\max \left\{1-\frac{|A_{(𝜀D)}|}{|G|}:A\subseteq G,\frac{|A|}{|G|}\ge \frac{1}{2}\right\}.$$

So $\alpha (G,𝜀)$ small says “$\frac{1}{2}$-sized sets have large $𝜀D$-neighbourhoods”.