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 $\beta$-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 $\text{mathds}{1}_{[a,b)}$.