The distribution of $d_4(n)$ in arithmetic progressions

We use the Petrow-Young [10] subconvexity bound for Dirichlet $L$-functions to show that $d_4(n)$ has exponent of distribution $4\mathrm{/}7$ when we allow an average over $a$ mod $q$, thereby giving an equidistribution result for $d_4(n)$ which goes past the $1\mathrm{/}2$ barrier for the first time.