Published online by Cambridge University Press: 19 June 2018
We prove that the exponent of distribution of $\unicode[STIX]{x1D70F}_{3}$ in arithmetic progressions can be as large as
$\frac{1}{2}+\frac{1}{34}$, provided that the moduli is squarefree and has only sufficiently small prime factors. The tools involve arithmetic exponent pairs for algebraic trace functions, as well as a double
$q$-analogue of the van der Corput method for smooth bilinear forms.