Expressive power of binary and ternary neural networks

We show that deep sparse ReLU networks with ternary weights and deep ReLU networks with binary weights can approximate $β$-Hölder functions on $[0,1]^d$. Also, for any interval $[a,b)\subset\mathbb{R}$, continuous functions on $[0,1]^d$ can be approximated by networks of depth $2$ with binary activation function $\mathds{1}_{[a,b)}$.