Khintchine inequalities and $Z_2$ sets

In this paper we show that the subset of integers that satisfies the Khintchine inequality for $p=1$ with the optimal constant $\sqrt{2}$ has to be a $Z_2$ set. We further prove a similar result for a large class of discrete groups. Our arguments rely on previous works by Haagerup/Musat \cite{Haagerup2007}, and Haagerup/Itoh \cite{Haagerup1995}.