Smooth numbers in arithmetic progressions to large moduli

We show that smooth numbers are equidistributed in arithmetic progressions to moduli of size $x^{66\mathrm{/}107-o(1)}$. This overcomes a longstanding barrier of $x^{3\mathrm{/}5-o(1)}$ present in previous works of Bombieri-Friedlander-Iwaniec, Fouvry-Tenenbaum, Drappeau, and Maynard. We build on Drappeau’s variation of the dispersion method and on exponential sum manipulations of Maynard, ultimately relying on optimized Deshouillers-Iwaniec type estimates for sums of Kloosterman sums.