Fourier series in BMO with number theoretical implications

This is a post-peer-review, pre-copyedit version of an article published in Mathematische Annalen. The final authenticated version is available online at: http://dx.doi.org/10.1007/s00208-019-01882-9We introduce an elementary argument to bound the 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 ∑n-1e2πin2x providing some insight into how much this BMO Fourier series differs from defining an L∞ functionThe authors are partially supported by the MTM2017-83496-P Grant of the MICINN (Spain) and the first author and the second author are also supported by “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554