  We determine the behavior of multiplicative functions vanishing at a positive
proportion of prime numbers in almost all short intervals. Furthermore we
quantify "almost all" with uniform power-saving upper bounds, that is, we save
a power of the suitably normalized length of the interval regardless of how
long or short the interval is. Such power-saving bounds are new even in the
special case of the M\"obius function. These general results are motivated by
several applications. First, we strengthen work of Hooley on sums of two
squares by establishing an asymptotic for the number of integers that are sums
of two squares in almost all short intervals. Previously only the order of
magnitude was known. Secondly, we extend this result to general norm forms of
an arbitrary number field $K$ (sums of two squares are norm-forms of
$\mathbb{Q}(i)$). Thirdly, Hooley determined the order of magnitude of the sum
of $(s_{n + 1} - s_{n})^{\gamma}$ with $\gamma \in (1, 5/3)$ where $s_{1} < s_2
< \ldots$ denote integers representable as sums of two squares. We establish a
similar results with $\gamma \in (1, 3/2)$ and $s_n$ the sequence of integers
representable as norm-forms of an arbitrary number field $K$. This is the first
such result for a number field of degree greater than two. Assuming the Riemann
Hypothesis for all Hecke $L$-functions we also show that $\gamma \in (1,2)$ is
admissible. Fourthly, we improve on a recent result of Heath-Brown about gaps
between $x^{\varepsilon}$-smooth numbers. More generally, we obtain results
about gaps between multiplicative sequences. Finally our result is useful in
other contexts aswell, for instance in our forthcoming work on Fourier
uniformity (joint with Terence Tao, Joni Terav\"ainen and Tamar Ziegler).
