On a density problem related to a theorem of Szegő

A classical theorem of Szegő states that for any probability measure $\mu =w\frac{\mathrm{d}\theta }{2\pi }+{\mu }_s$ on the unit circle the polynomials are dense in $L^2(\mathbb{T},\mu )$ if and only if $\log (w)\mathrm{\notin }{L}^1(\mathbb{T})$. A related question asks whether the monomials with exponents in some subset $\mathrm{\Lambda }\subseteq {\mathbb{N}}_0$ already span $L^2(\mathbb{T},\mu )$ if $\log (w)\mathrm{\notin }{L}^1(\mathbb{T})$. A result by Olevskii and Ulanovskii gives an answer if $\mu$ belongs to a class of absolutely continuous measures. We investigate the same question for Markoff measures.