Fourier series in BMO with number theoretical implications

Abstract

We introduce an elementary argument to bound the $\text{BMO}$ 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 $\sum n^{-1}{e}^{2\pi i{n}^{2}x}$ providing some insight into how much this $\text{BMO}$ Fourier series differs from defining an $L^{\infty }$ function.

15 pages, no figures. Submitted

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