An information scaling law: ζ= 3/4

Consider the entropy of a unit Gaussian convolved over a discrete set of K points, constrained to an interval of length L. Maximising this entropy fixes K, and we show that this number exhibits a novel scaling law K ~ L^1/\zeta as L -> infinity, with exponent \zeta = 3/4. This law was observed numerically in a recent paper about optimal effective theories; here we present an analytic derivation. We argue that this law is generic for channel capacity maximisation, or the equivalent minimax problem. We also briefly discuss the behaviour at the boundary of the interval, and higher dimensional versions.