Spectral clumping for functions decreasing rapidly on a half-line

We demonstrate a phenomenon of condensation of the Fourier transform $\widehat{f}$ of a function $f$ defined on the real line $\mathbb{R}$ which decreases rapidly on one half of the line. For instance, we prove that if $f$ is square-integrable on $\mathbb{R}$, then a one-sided estimate of the form $$\rho_f(x) := \int_x^{\infty} |f(t)| \,dt = \mathcal{O}\big(e^{-c\sqrt{x}} \big), \quad x > 0$$ for some $c > 0$, forces the non-zero frequencies $\sigma(f) := \{ \zeta \in \mathbb{R} : |\widehat{f}(\zeta)| > 0 \}$ to clump: this set differs from an open set $U$ only by a set of Lebesgue measure zero, and $\log |\widehat{f}|$ is locally integrable on $U$. In particular, if $f$ is non-zero, then there exists an interval on which $\log |\widehat{f}|$ is integrable. The roles of $f$ and $\widehat{f}$ above may be interchanged, and the result extends also to a large class of tempered distributions. We show that the above decay condition is close to optimal, in the following sense: a non-zero entire function $f$ exists which is square-integrable on $\mathbb{R}$, for which $\sigma(f)$ is a subset of a compact set $E$ containing no intervals, and for which the estimate $\rho_f(x) = \mathcal{O}\big( e^{-x^a}\big)$, $x > 0$, holds for every $a \in (0, 1/2)$.