On asymptotics of shifted sums of Dirichlet convolutions

The objective of this paper is to obtain asymptotic results for shifted sums of multiplicative functions of the form $g\ast 1$, where the function $g$ satisfies the Ramanujan conjecture and has conjectured upper bounds on square moments of its L-function. We establish that for $H$ within the range $X^{23\mathrm{/}24+10\epsilon }\le H\le X^{1-\epsilon }$, there exist constants $B_{f,h}$ such that $$\sum _{X\le n\le 2X}f(n)f(n+h)-B_{f,h}X=O_{f,\epsilon }\left(X^{1-{\epsilon }^2\mathrm{/}4}\right)$$ for all but $O_{f,\epsilon }\left(H{X}^{-{\epsilon }^2\mathrm{/}3}\right)$ integers $h\in [1,H]\mathrm{.}$ Our method is based on the Hardy-Littlewood circle method. In order to treat minor arcs, we use the convolution structure and a cancellation of $g(n)$ that are additively twisted, applying some arguments from a paper of Matomaki, Radziwill and Tao. Also, we establish an upper bound for weighted exponential sums, which may be of independent interest.