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

Abstract:

Suppose $D$ is a suitably admissible compact subset of $\mathbb {R}^k$ having a smooth boundary with possible zones of zero curvature. Let $R(T,\theta ,x)= N(T,\theta ,x) - T^{k}\mathrm {vol}(D)$, 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 $(1/T)\int _{1}^{T} |R(t,\theta ,x) dt| \ll T^{\frac {1}{4} +\epsilon }$ is valid with an improved estimate. We conclude with an account of hyperbolic versions for which, drawing on previous work of Hill and Parnovski, we give counterparts in all dimensions for both the compact and non-compact finite volume cases.