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

Fourier series in BMO with number theoretical implications