Infinite necklace: the line as a circle

Represent the line by the real numbers. By bijection, that’s equivalent to [0,1], the closed unit interval. Now cut it into an unlimited (infinitely large) number of pieces. Name that number of pieces “H”. Then [0,1] ~ Z* mod H where Z* is the hyperintegers and H is hyperfinite. This gives a useful intermediate object that captures things like a point at infinity in projective spaces (instead of nilsquare vectors, you can just designate some basis vectors infinitesimal length).

Let’s use it to link the discrete and continuous Fourier transform. The continuous is the standard part aka shadow of a hyperdiscrete Fourier transform so the roots of unity are sampled as i/H.

This is all brutally rough so feel free to request elaboration.