I remember from a long time ago, seeing an way of getting low power
    from thin air. You need a circuit resonant at the frequency of your
    local AM transmitter, an antenna and earth and a rectifier tapped into
    the coil.

Yep.  This is exactly how a "crystal radio" (no batteries, no power
supply) works.  I have/had somewhere a book that used two tuners - one
tuned to a strong local station that acted as a power supply for a
more conventional amplified tuner...

BillW