Level of distribution of unbalanced convolutions

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.