Certified rational nets and a budgeted mesh theorem

For fixed $s$, the certified radii tend to zero as $N$ grows. For example, taking denominator $2^h$ with $h=\lfloor (N-s)/2\rfloor$ gives a fine-graining grid whose rounding radius is at most

The argument of ${\omega }_N$ is the square root of this probability radius. There is no strategy-uniform rate without a common modulus assumption.