paulfchristiano comments on Paper: “A simple function of one parameter (θ) can fit any collection of ordered pairs {Xi,Yi} to arbitrary precision”—implications for Occam?

[paulfchristiano]

Sorry, we need sin(ax + b).

[Dacyn]

Still doesn’t work. If $(x_i{)}_{i=1}^3$ is an arithmetic progression then $\left(\exp \right(i(a{x}_i+b)){)}_{i=1}^3$ is a geometric progression, the imaginary part of which is $\left(\sin \right(a{x}_i+b){)}_{i=1}^3$. This implies that for example $\sin (ax+b)$ cannot simultaneously approximate $(0,0)$, $(1,0)$, and $(2,1)$.