On the Rankin-Selberg problem in short intervals

If $$\mathrm{\Delta }(x):=\sum _{n⩽x}{c}_n-Cx(C>0)$$ denotes the error term in the classical Rankin-Selberg problem, then we obtain a non-trivial upper bound for the mean square of $\mathrm{\Delta }(x+U)-\mathrm{\Delta }(x)$ for a certain range of $U=U(X)$. In particular, under the Lindel"of hypothesis for $\zeta (s)$, it is shown that $${\int }_X^{2X}(\mathrm{\Delta }(x+U)-\mathrm{\Delta }(x){)}^2\text{roman}dx{\ll }_{ϵ}{X}^{9\mathrm{/}7+ϵ}{U}^{8\mathrm{/}7},$$ while under the Lindel"of hypothesis for the Rankin-Selberg zeta-function the integral is bounded by $X^{1+ϵ}{U}^{4\mathrm{/}3}$. An analogous result for the discrete second moment of $\mathrm{\Delta }(x+U)-\mathrm{\Delta }(x)$ also holds.