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)

 
 

 

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


Author: Burton Randol
Journal: Trans. Amer. Math. Soc. 370 (2018), 3113-3127
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?)

  • [1] P. Bleher and J. Bourgain, Distribution of the error term for the number of lattice points inside a shifted ball, Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) Progr. Math., vol. 138, Birkhäuser Boston, Boston, MA, 1996, pp. 141-153. MR 1399335, https://doi.org/10.1007/s10107-012-0541-z
  • [2] Pavel M. Bleher, Zheming Cheng, Freeman J. Dyson, and Joel L. Lebowitz, Distribution of the error term for the number of lattice points inside a shifted circle, Comm. Math. Phys. 154 (1993), no. 3, 433-469. MR 1224087
  • [3] L. Brandolini, S. Hofmann, and A. Iosevich, Sharp rate of average decay of the Fourier transform of a bounded set, Geom. Funct. Anal. 13 (2003), no. 4, 671-680. MR 2006553, https://doi.org/10.1007/s00039-003-0426-7
  • [4] Peter Buser, Riemannsche Flächen mit Eigenwerten in $ (0,$ $ 1/4)$, Comment. Math. Helv. 52 (1977), no. 1, 25-34. MR 0434961, https://doi.org/10.1007/BF02567355
  • [5] K. Chandrasekharan and Raghavan Narasimhan, On lattice-points in a random sphere, Bull. Amer. Math. Soc. 73 (1967), 68-71. MR 0211967, https://doi.org/10.1090/S0002-9904-1967-11644-6
  • [6] Harald Cramér, Über zwei Sätze des Herrn G. H. Hardy, Math. Z. 15 (1922), no. 1, 201-210 (German). MR 1544568, https://doi.org/10.1007/BF01494394
  • [7] Y. Colin de Verdière, Nombre de points entiers dans une famille homothétique de domains de $ {\bf R}$, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 4, 559-575 (French). MR 0480399
  • [8] Jean Delsarte, Sur le gitter fuchsien, C. R. Acad. Sci. Paris 214 (1942), 147-179 (French). MR 0007769
  • [9] Jean Delsarte, Le gitter fuchsien, Oeuvres de Jean Delsarte, t. II, Éditions du CNRS, Paris, 1971, pp. 829-845.
  • [10] G. H. Hardy, The average order of the arithmetical functions $ P(x)$ and $ \delta(x)$, Proc. London Math. Soc. S2-15, no. 1, 192. MR 1576556, https://doi.org/10.1112/plms/s2-15.1.192
  • [11] Dennis A. Hejhal, The Selberg trace formula for $ {\rm PSL}(2,R)$. Vol. I, Lecture Notes in Mathematics, Vol. 548, Springer-Verlag, Berlin-New York, 1976. MR 0439755
  • [12] R. Hill and L. Parnovski, The variance of the hyperbolic lattice point counting function, Russ. J. Math. Phys. 12 (2005), no. 4, 472-482. MR 2201311
  • [13] Heinz Huber, Über eine neue Klasse automorpher Funktionen und ein Gitterpunktproblem in der hyperbolischen Ebene. I, Comment. Math. Helv. 30 (1956), 20-62 (1955) (German). MR 0074536, https://doi.org/10.1007/BF02564331
  • [14] Alex Iosevich and Elijah Liflyand, Decay of the Fourier transform: Analytic and geometric aspects, Birkhäuser/Springer, Basel, 2014. MR 3308120
  • [15] Henryk Iwaniec, Spectral methods of automorphic forms, 2nd ed., Graduate Studies in Mathematics, vol. 53, American Mathematical Society, Providence, RI; Revista Matemática Iberoamericana, Madrid, 2002. MR 1942691
  • [16] David G. Kendall, On the number of lattice points inside a random oval, Quart. J. Math., Oxford Ser. 19 (1948), 1-26. MR 0024929
  • [17] Hugues Lapointe, Iosif Polterovich, and Yuri Safarov, Average growth of the spectral function on a Riemannian manifold, Comm. Partial Differential Equations 34 (2009), no. 4-6, 581-615. MR 2530710, https://doi.org/10.1080/03605300802537453
  • [18] Peter D. Lax and Ralph S. Phillips, The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces, J. Funct. Anal. 46 (1982), no. 3, 280-350. MR 661875, https://doi.org/10.1016/0022-1236(82)90050-7
  • [19] B. M. Levitan, Asymptotic formulas for the number of lattice points in Euclidean and Lobachevskiĭ spaces, Uspekhi Mat. Nauk 42 (1987), no. 3(255), 13-38, 255 (Russian). MR 896876
  • [20] W. Luo, Z. Rudnick, and P. Sarnak, On Selberg's eigenvalue conjecture, Geom. Funct. Anal. 5 (1995), no. 2, 387-401. MR 1334872, https://doi.org/10.1007/BF01895672
  • [21] John J. Millson, On the first Betti number of a constant negatively curved manifold, Ann. of Math. (2) 104 (1976), no. 2, 235-247. MR 0422501, https://doi.org/10.2307/1971046
  • [22] Werner Müller, Spectral theory of automorphic forms, http://www.math.uni-bonn.de/people/mueller/skripte/specauto.pdf (2010), 1-61.
  • [23] Jean-Pierre Otal and Eulalio Rosas, Pour toute surface hyperbolique de genre $ g,\ \lambda_{2g-2}>1/4$, Duke Math. J. 150 (2009), no. 1, 101-115 (French, with English and French summaries). MR 2560109, https://doi.org/10.1215/00127094-2009-048
  • [24] S. J. Patterson, A lattice-point problem in hyperbolic space, Mathematika 22 (1975), no. 1, 81-88. MR 0422160, https://doi.org/10.1112/S0025579300004526
  • [25] Ralph Phillips and Zeév Rudnick, The circle problem in the hyperbolic plane, J. Funct. Anal. 121 (1994), no. 1, 78-116. MR 1270589, https://doi.org/10.1006/jfan.1994.1045
  • [26] Burton Randol, A lattice-point problem, Trans. Amer. Math. Soc. 121 (1966), 257-268. MR 0201407, https://doi.org/10.2307/1994344
  • [27] Burton Randol, A lattice-point problem. II, Trans. Amer. Math. Soc. 125 (1966), 101-113. MR 0201408, https://doi.org/10.2307/1994590
  • [28] Burton Randol, On the Fourier transform of the indicator function of a planar set, Trans. Amer. Math. Soc. 139 (1969), 271-278. MR 0251449, https://doi.org/10.2307/1995319
  • [29] Burton Randol, On the asymptotic behavior of the Fourier transform of the indicator function of a convex set, Trans. Amer. Math. Soc. 139 (1969), 279-285. MR 0251450, https://doi.org/10.2307/1995320
  • [30] Burton Randol, Small eigenvalues of the Laplace operator on compact Riemann surfaces, Bull. Amer. Math. Soc. 80 (1974), 996-1000. MR 0400316, https://doi.org/10.1090/S0002-9904-1974-13609-8
  • [31] Burton Randol, The Riemann hypothesis for Selberg's zeta-function and the asymptotic behavior of eigenvalues of the Laplace operator, Trans. Amer. Math. Soc. 236 (1978), 209-223. MR 0472728, https://doi.org/10.2307/1997781
  • [32] Burton Randol, A Dirichlet series of eigenvalue type with applications to asymptotic estimates, Bull. London Math. Soc. 13 (1981), no. 4, 309-315. MR 620043, https://doi.org/10.1112/blms/13.4.309
  • [33] Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
  • [34] Ingvar Svensson, Estimates for the Fourier transform of the characteristic function of a convex set, Ark. Mat. 9 (1971), 11-22. MR 0328471, https://doi.org/10.1007/BF02383634
  • [35] A. N. Varchenko, On the number of lattice points in a domain, Uspekhi Mat. Nauk 37 (1982), no. 3(225), 177-178 (Russian). MR 659434
  • [36] A. N. Varchenko, The number of lattice points in families of homothetic domains in $ {\bf R}^{n}$, Funktsional. Anal. i Prilozhen. 17 (1983), no. 2, 1-6 (Russian). MR 705041
  • [37] William Wolfe, The asymptotic distribution of lattice points in hyperbolic space, J. Funct. Anal. 31 (1979), no. 3, 333-340. MR 531135, https://doi.org/10.1016/0022-1236(79)90007-7
  • [38] A. A. Yudin, On the number of integer points in the displaced circles, Acta Arith. 14 (1967/1968), 141-152. MR 0229601

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

American Mathematical Society