2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/103853A 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.25 pages. Part of GK's PhD thesisClassical Analysis and ODEsComplex Variables42b20text