There is no universal Riemann–Lebesgue decay rate

Set ${\xi }_0=0$. For $k\ge 1$ let ${\xi }_k$ be the least positive integer greater than ${\xi }_{k-1}+k$ for which $r({\xi }_k)<2^{-2k}$. Such integers exist by the assumed limit. This recursion is explicit and uses no choice selection. The series $f(x)={\sum }_{k\ge 1}{2}^{-k}{e}^{2\pi i{\xi }_{k}x}g(x)$ converges absolutely at each x with modulus at most g(x); its measurable limit belongs to $L^1$ by F1.

At each frequency, F3 applied to the partial sums times the unit-modulus Fourier factor, dominated by g, gives $\widehat{f}(\xi )={\sum }_{k\ge 1}{2}^{-k}{e}^{-\pi (\xi -{\xi }_k{)}^2}$ by F1 and F2. All summands are nonnegative. Thus $\mathrm{\mid }\widehat{f}({\xi }_k)\mathrm{\mid }\ge 2^{-k}$ and $\mathrm{\mid }\widehat{f}({\xi }_k)\mathrm{\mid }\mathrm{/}r({\xi }_k)>2^k\to \mathrm{\infty }$, while ${\xi }_k\to \mathrm{\infty }$. This refutes every proposed rate, even allowing an f-dependent big-O constant. Countable choice is inherited only from the Gaussian and modulation suppliers. This explicit construction is local and is not attributed to a source theorem.