Short Exponential Sums and Ternary Correlations of Multiplicative Functions

Let $f_1, f_2, f_3$ be $k$-divisor-bounded functions, at least one of which satisfies certain second-moment integral bounds. We show that for any $\varepsilon > 0$ and \[ X^{1/2+100\varepsilon} \ll H \ll X^{1-\varepsilon}, \] we have \[ \sum_{|h|\le H} \left(1-\frac{|h|}{H}\right) \sum_{X \le n \le 2X} f_{1}(n)\, f_{2}(n+h)\, f_{3}(n+2h) = O\!\left(XH^{1-\varepsilon}\right). \] Our approach differs from previous methods based on spectral theory or Heath-Brown-type decompositions, and instead combines the circle method with weighted short exponential-sum bounds. The key input is short exponential-sum estimates obtained from integral moment bounds for $L$-functions.