Fourier series in BMO with number theoretical implications

We introduce an elementary argument to bound the $\textrm{BMO}$ seminorm of Fourier series with gaps giving in particular a sufficient condition for them to be in this space. Using finer techniques we carry out a detailed study of the series $\sum n^{-1}e^{2πi n^2 x}$ providing some insight into how much this $\text{BMO}$ Fourier series differs from defining an $L^\infty$ function.