Haar Equidistribution in the Actual Torus Orbit Closure
Abstract
Every continuous observable on a finite torus orbit closure has its normalized Haar orbit average.
Theorem 1.1 (All continuous observables on the closed cyclic subgroup).
Proof. Machine-checked in Lean as D5/S3/Fourier/Asymptotics/TorusOrbitEquidistribution.result (✓ std3). ∎
Citation. Terence Tao (2008). The van der Corput trick, and equidistribution on nilmanifolds. URL: https://terrytao.wordpress.com/2008/06/14/the-van-der-corputs-trick-and-equidistribution-on-nilmanifolds/.
Commentary.
Let I be any finite index type, including the empty type, and let g be any element of the multiplicative torus T^I, where T is the complex unit circle. G is the topological closure of the cyclic subgroup generated by g. The first equality identifies its carrier with the closure of the nonnegative powers. The measure mu_G is Haar measure on G normalized to total mass one.
The limit holds for every continuous complex-valued function on G. The sum samples exactly g through g^N. At N=0 the sum and its totalized quotient are zero; this convention has no effect on the limit. No irrationality, independence, ambient density or connectedness is required.
For each integer character, split according to whether its value at g is one. In that case it equals one on all powers and, by continuity, on G. Otherwise the finite geometric sum divided by N tends to zero. Translation invariance of Haar measure forces the same character’s integral to vanish. This split retains every integer relation among the coordinates.
Complex Tietze extension supplies an ambient continuous extension for every observable on G. Restricting the dense multivariate Fourier span therefore gives a dense subspace of C(G,C). The averaging operators have norm at most one and Haar integration is continuous, so the observables satisfying the limit form a closed complex subspace containing that dense span.
This is the classical character proof of compact-group rotation equidistribution. The cited exposition states the ambient Weyl criterion and geometric-series argument and illustrates a proper subtorus; the restriction and Haar-invariance steps here give the complete orbit-closure form. Finite-order and disconnected groups are included.