Support and Robust Returns of Torus Observations
Abstract
Vanishing errors preserve bounded returns to every open window meeting a continuous torus observation image.
Theorem 1.1 (Full image support, bounded waiting and positive lower density).
Proof. Machine-checked in Lean as D5/S3/Fourier/Asymptotics/TorusObservationWindowReturns.result (✓ std3). ∎
Source. Repository-derived.
Acknowledgement. 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 type, T the complex unit circle, and g an element of T^I. Let G be the closure of the cyclic subgroup generated by g and let mu be its Haar probability measure. The observable h is any continuous real function on G, and e is any real sequence whose difference from h(g^n) tends to zero. Write nu for the pushforward of mu by h. For J=(u,v), let A_J(N) count the integers m with 0 <= m < N and e_m in J.
The support of nu is exactly h(G). For every open interval J meeting this image, the same integers n_0 >= 1 and M >= 0 give both conclusions: each interval of integers from n to n+M contains a hit whenever n >= n_0, and the lower asymptotic density of the hits is at least 1/(M+1). The last displayed clause states this density bound as an eventual epsilon inequality. The counting ratio lies in [0,1], so this is equivalent to the usual liminf inequality. Counting from 1 instead of 0 changes a ratio by at most 1/N and gives the same lower density.
Compactness makes h(G) closed. An open neighborhood of any point h(z) has a nonempty open preimage, which has positive Haar measure. The complement of h(G) has empty preimage. These two facts identify the support, including atomic pushforward measures.
Choose z with h(z) in J and a positive margin delta so that the interval from h(z)-2 delta to h(z)+2 delta lies inside J. The inverse image U of the smaller interval with radius delta is nonempty and open. Nonnegative powers are dense in the actual closed subgroup, so every translate x g^k enters U for some k >= 0. The corresponding open preimages cover G. A finite subcover supplies a maximum time M, valid for every starting point. Eventually the perturbation has absolute value less than delta, preserving each selected hit in J.
Put L=M+1 and divide the integers from n_0 onward into disjoint blocks of length L. Choose one hit per block. The chosen times increase strictly, hence the first floor((N-n_0)/L) complete blocks give that many distinct hits below N when N >= n_0. It follows that N <= n_0 + L(A_J(N)+1). Dividing by N L and letting N grow proves the claimed eventual epsilon inequalities.
The statement includes empty I, finite-order generators, proper or disconnected subgroups, constant observables, and zero or nonzero vanishing perturbations. It requires neither ambient density nor atom-free thresholds, and assumes no rate of convergence. The waiting bound depends on the generator, observable and window; no bound uniform over these parameters is asserted. The cited torus equidistribution exposition supplies background on orbit closures; the support and finite-cover argument here is a synthesis of standard compact-group and counting methods.