A Hardy-type result 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 \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 $E_{\ast }^{+}(k)$, where $E_{\ast }^{+}(k)$ is the quotient of the direct Euclidean group by the subgroup of integral translations, and $E_{\ast }^{+}(k)$ 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 $(\theta ,x)\in E_{\ast }^{+}(k)$, one consequence of which in two dimensions is that for almost all $(\theta ,x)\in E_{\ast }^{+}(2)$, 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.

arXiv commentThis version generalizes the domains in $R^k$ to which the results apply, and adds additional references