Bilinear sums with $GL(2)$ coefficients and the exponent of distribution of $d_3$

We obtain the exponent of distribution $1\mathrm{/}2+1\mathrm{/}30$ for the ternary divisor function $d_3$ to square-free and prime power moduli, improving the previous results of Fouvry–Kowalski–Michel, Heath-Brown, and Friedlander–Iwaniec. The key input is certain estimates on bilinear sums with $GL(2)$ coefficients obtained using the delta symbol approach.