  It is proved that, if $k\ge2$ is a fixed integer and $1 \ll H \le X/2$, then
$$ \int_{X-H}^{X+H}\Delta^4_k(x)\d x \ll_\epsilon X^\epsilon\Bigl(HX^{(2k-2)/k}
+ H^{(2k-3)/(2k+1)}X^{(8k-8)/(2k+1)}\Bigr), $$ where $\Delta_k(x)$ is the error
term in the general Dirichlet divisor problem. The proof uses the
Vorono{\"\i}--type formula for $\Delta_k(x)$, and the bound of Robert--Sargos
for the number of integers when the difference of four $k$--th roots is small.
We also investigate the size of the error term in the asymptotic formula for
the $m$-th moment of $\Delta_2(x)$.
