I encountered this proposition: " $\int\limits_{-\infty}^{\infty} |f(t)| dt < \infty$, i.e. $f(t)$ has to grow slower than an exponential curve."
Is exponential growth the slowest increment that gives a nonintegrable function (in the sense of Lebesgue-integral)?
Although rather imprecise, let me remark what follows, where functions are positive fpr simplicity. Asssume $h \notin L^1(\mathbb{R}^N)$. If $f\in L^1(\mathbb{R}^N)$, then $$ \limsup_{|x| \to +\infty} \frac{f(x)}{h(x)} =0. $$ Otherwise, for some constant $C>0$ we would have eventually $f(x) \geq C h(x)$. and therefore $\int_{\mathbb{R}^N} f(x) \, dx = +\infty$.
However, I guess that there is no precise threshold for integrability, like there is no threshold for the convergence of a numerical series.