A Hardy-type result on the average of the lattice point error term over long intervals

Suppose $D$ is a suitably admissible compact subset of $\mathbb{R}^k$ having a smooth boundary with possible zones of zero curvature. Let \mbox{$R(T,\theta,x)= N(T,\theta,x) - T^{k}\mathrm{vol}(D)$,} where $N(T,\theta,x)$ is the number of integral lattice points contained in an $x$-translation of $T\theta(D)$, with $T >0$ a dilation parameter and $\theta \in SO(k)$. Then $R(T,\theta,x)$ can be regarded as a function with parameter $T$ on the space $E_{*}^{+}(k)$, where $E_{*}^{+}(k)$ is the quotient of the direct Euclidean group by the subgroup of integral translations, and $E_{*}^{+}(k)$ has a normalized invariant measure which is the product of normalized measures on $SO(k)$ and the $k$-torus. We derive an integral estimate, valid for almost all $(\theta,x) \in E_{*}^{+}(k)$, one consequence of which in two dimensions is that for almost all $(\theta,x) \in E_{*}^{+}(2)$, a counterpart of the Hardy circle estimate \mbox{$(1/T)\int_{1}^{T} |R(t,\theta,x)\,dt| \ll T^{\frac{1}{4} +\epsilon}\;$}is valid with an improved estimate. We conclude with an account of hyperbolic versions for which, drawing on previous work of Hill and Parnovski \cite{hill-parnovski}, we give counterparts in all dimensions, for both the compact and non-compact finite volume cases.