Neural networks with superexpressive activations and integer weights

An example of an activation function $σ$ is given such that networks with activations $\{σ, \lfloor\cdot\rfloor\}$, integer weights and a fixed architecture depending on $d$ approximate continuous functions on $[0,1]^d$. The range of integer weights required for $\varepsilon$-approximation of Hölder continuous functions is derived, which leads to a convergence rate of order $n^{\frac{-2β}{2β+d}}\log_2n$ for neural network regression estimation of unknown $β$-Hölder continuous function with given $n$ samples.