A constraint on extensible quadrature rules

When the worst case integration error in a family of functions decays as $n^{-\alpha }$ for some $\alpha >1$ and simple averages along an extensible sequence match that rate at a set of sample sizes $n_1<n_2<\cdots <\mathrm{\infty }$, then these sample sizes must grow at least geometrically. More precisely, $n_{k+1}\mathrm{/}{n}_k\ge \rho$ must hold for a value $1<\rho <2$ that increases with $\alpha$. This result always rules out arithmetic sequences but never rules out sample size doubling. The same constraint holds in a root mean square setting.