  We show that if an essentially arbitrary sequence supported on an interval
containing $x$ integers, is convolved with a tiny Siegel-Walfisz-type sequence
supported on an interval containing $\exp((\log x)^{\varepsilon})$ integers
then the resulting multiplicative convolution has (in a weak sense) level of
distribution $x^{1/2 + 1/66 - \varepsilon}$ as $x$ goes to infinity. This
dispersion estimate has a number of consequences for: the distribution of the
$k$th divisor function to moduli $x^{1/2 + 1/66 - \varepsilon}$ for any integer
$k \geq 1$, the distribution of products of exactly two primes in arithmetic
progressions to large moduli, the distribution of sieve weights of level
$x^{1/2 + 1/66 - \varepsilon}$ to moduli as large as $x^{1 - \varepsilon}$ and
for the Brun-Titchmarsh theorem for almost all moduli $q$ of size $x^{1 -
\varepsilon}$, lowering the long-standing constant $4$ in that range. Our
result improves and is inspired by earlier work of Green (and subsequent work
of Granville-Shao) which is concerned with the distribution of $1$-bounded
multiplicative functions in arithmetic progressions to large moduli. As in
these previous works the main technical ingredient are the recent estimates of
Bettin-Chandee for trilinear forms in Kloosterman fractions and the estimates
of Duke-Friedlander-Iwaniec for bilinear forms in Kloosterman fractions.
