# Prove the result for shifted HAPs instead of HAPs

For an explanation of the justification for considering shifted HAPs, or privileged APs (PAPs), see this post. The simplest way to generate bounded-discrepancy PAPs experimentally is to choose a very large number $N$ at random (the shift), calculate $r_d$ for all small $d$ such that $d \mid N + r_d$ where $0 \leq r_d \lt d$, and then construct a zero-based sequence $(x_n)$ requiring that $| x_{r_d} + x_{r_d + d} + x_{r_d + 2d} + \ldots + x_{r_d + md} | \leq C$ for all $d$ and $m$.