  We introduce a Littlewood-Paley characterization of modulation spaces and use
it to give an alternative proof of the algebra property, somehow implicitly
contained in Sugimoto (2011), of the intersection $M^s_{p,q}(\mathbb{R}^d) \cap
M_{\infty, 1}(\mathbb{R}^d)$ for $d \in \mathbb{N}$, $p, q \in [1, \infty]$ and
$s \geq 0$. We employ this algebra property to show the local well-posedness of
the Cauchy problem for the cubic nonlinear Schr\"odinger equation in the above
intersection. This improves Theorem 1.1 by B\'enyi and Okoudjou (2009), where
only the case $q = 1$ is considered, and closes a gap in the literature. If $q
> 1$ and $s > d \left(1 - \frac{1}{q}\right)$ or if $q = 1$ and $s \geq 0$ then
$M^s_{p,q}(\mathbb{R}^d) \hookrightarrow M_{\infty, 1}(\mathbb{R}^d)$ and the
above intersection is superfluous. For this case we also reobtain a
H\"older-type inequality for modulation spaces.
