TITLE: Integrals of the form $\int_0^{+\infty} \sin g(x) \ dx$
QUESTION [15 upvotes]: I'm interested in the convergence of integrals of the form $$\int_0^{+\infty} \sin g(x) \ dx$$
where $g$ is nonnegative, increasing and growing without bound as $x \to +\infty$ (hence $\sin g(x)$ oscillates as $x \to +\infty$). For example, it's known that $$\int_0^{+\infty} \sin x^p \ dx$$ converges for $p>1$. Similarly, $\int_0^{+\infty} \sin(\exp x) \ dx$ converges. 
Here are three questions (they are related enough that I didn't think it was worth making three separate questions). 

Consider functions $g:[0, +\infty) \to [0, +\infty)$ which are continuous, strictly increasing and unbounded. 
$(i)$ Suppose we add the additional hypothesis that $g$ is strictly convex. Does $\int_0^{+\infty} \sin g(x) \ dx$ converge? 
$(ii)$ Can we characterize those $g$'s for which $\int_0^{+\infty} \sin g(x) \ dx$ converges?
$(iii)$ Is there a $g$ such that $\int_0^{+\infty} \sin g(x) \ dx$ converges absolutely?

REPLY [2 votes]: For (iii).
Let $(c_n)$ be a sequence of positive numbers, $0 < c_n < \pi$ for every $n$, and let
$$
h(x) := \sum_{n=0}^\infty \frac{\pi}{c_n} \chi_{[n\pi, n\pi + c_n]}(x),
\qquad
g(x) := \int_0^x h(t)\, dt,
$$
where $\chi_A$ is the characteristic function of the set $A$.
(This function $g$ is not strictly increasing, but the construction can be easily modified in this sense.)
Since $\int_{n\pi}^{(n+1)\pi} g'(x)\, dx = \pi$, we have that $g(n\pi) = n\pi$
and
$$
\int_{n\pi}^{(n+1)\pi} |\sin g(x)|\, dx 
= \int_0^{c_n} \sin\left(\frac{\pi}{c_n}\, t\right)\, dt = \frac{2 c_n}{\pi}.
$$
Hence, if $\sum_n c_n$ converges, then $\sin g(x)$ is absolutely integrable on $[0,+\infty)$.