Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 
 
 

 

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


Author: Burton Randol
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11F72, 20F69, 11P21, 35P20
DOI: https://doi.org/10.1090/tran/7043
Published electronically: October 31, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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} \vert R(t,\theta ,x)\,dt\vert \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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11F72, 20F69, 11P21, 35P20

Retrieve articles in all journals with MSC (2010): 11F72, 20F69, 11P21, 35P20


Additional Information

Burton Randol
Affiliation: Ph.D. Program in Mathematics, Graduate Center of CUNY, 365 Fifth Avenue, New York, New York 10016

DOI: https://doi.org/10.1090/tran/7043
Received by editor(s): March 22, 2016
Received by editor(s) in revised form: July 20, 2016
Published electronically: October 31, 2017
Article copyright: © Copyright 2017 American Mathematical Society