  A Menshov spectrum is a subset of the integers that is sufficient for
representing every measurable function as an almost-everywhere converging
trigonometric (non-Fourier) sum. In this language the celebrated "Menshov
representation theorem" states that Z is a Menshov spectrum. In this paper we
construct 1) Menshov spectra that are almost exponentially sparse 2) that are
almost squares. Then we show that the positive integers are not a Menshov
spectrum but are a Menshov spectrum in measure, and repeat 1) and 2) in the
analytic settings.
