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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Hardy-type results on the average of the lattice point error term over long intervals
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by Burton Randol PDF
Trans. Amer. Math. Soc. 370 (2018), 3113-3127 Request permission


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.
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Additional Information
  • Burton Randol
  • Affiliation: Ph.D. Program in Mathematics, Graduate Center of CUNY, 365 Fifth Avenue, New York, New York 10016
  • MR Author ID: 211692
  • Received by editor(s): March 22, 2016
  • Received by editor(s) in revised form: July 20, 2016
  • Published electronically: October 31, 2017
  • © Copyright 2017 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 3113-3127
  • MSC (2010): Primary 11F72, 20F69, 11P21, 35P20
  • DOI:
  • MathSciNet review: 3766843