We look at a finite collection of time-frequency shifts built from two parts: Lattice part – all but one shift lie on the usual integer grid; Rogue shift – one extra shift sits off that grid. For any non-zero Schwartz function f, the vectors obtained by applying those shifts to f are linearly independent. Assuming a dependence leads, via the Zak transform, to a torus equation tying size and phase. An orbit trichotomy for the induced translation then shows: dense orbits force f = 0; finite orbits invoke Linnell’s lattice theorem; infinite but non-dense orbits are excluded by a new rigidity argument blending ergodic averages with phase analysis. These three mutually exclusive cases cover all possibilities, settling the Heil–Ramanathan–Topiwala conjecture for mixed-integer configurations with Schwartz windows.
Session
AMS Special Session on Harmonic Analysis, Frame Theory, and Tilings I