TY - JOUR

T1 - The circle problem in the hyperbolic plane

AU - Phillips, Ralph

AU - Rudnick, Zeév

PY - 1994/4

Y1 - 1994/4

N2 - This paper deal with the analogue of the classical circle problem in the hyperbolic plane; that is, we count the number NΓ(s, z) of translates of a base point z by a Fuchsian group Γ, which lie in a geodesic ball of radius s about z. If Σ(s, z) is the theoretical best approximation to N(s, z) (which has πes/vol(Γ) as its leading term), we set d(s, z) = N(s, z) - Σ(s, z), the best known upper bound for which is O(e2s/3). We get omega results for d(s, z), which in the co-compact case are d(s, z) = Ω(es/2β(s)), where β(s) → ∞ as s → ∞. We also show that the normalized remainder term e(s, z) = d(s, z)/es/2 has finite mean, which is zero unless Γ is noncompact and has null forms. Further we carry out a numerical investigation of the Fermat groups and the results are consistent with an upper bound e(s, z) = O(eε(lunate)S) for all ε(lunate) > 0. The problem in hyperbolic n-space is also investigated.

AB - This paper deal with the analogue of the classical circle problem in the hyperbolic plane; that is, we count the number NΓ(s, z) of translates of a base point z by a Fuchsian group Γ, which lie in a geodesic ball of radius s about z. If Σ(s, z) is the theoretical best approximation to N(s, z) (which has πes/vol(Γ) as its leading term), we set d(s, z) = N(s, z) - Σ(s, z), the best known upper bound for which is O(e2s/3). We get omega results for d(s, z), which in the co-compact case are d(s, z) = Ω(es/2β(s)), where β(s) → ∞ as s → ∞. We also show that the normalized remainder term e(s, z) = d(s, z)/es/2 has finite mean, which is zero unless Γ is noncompact and has null forms. Further we carry out a numerical investigation of the Fermat groups and the results are consistent with an upper bound e(s, z) = O(eε(lunate)S) for all ε(lunate) > 0. The problem in hyperbolic n-space is also investigated.

UR - http://www.scopus.com/inward/record.url?scp=0003095580&partnerID=8YFLogxK

U2 - 10.1006/jfan.1994.1045

DO - 10.1006/jfan.1994.1045

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:0003095580

SN - 0022-1236

VL - 121

SP - 78

EP - 116

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 1

ER -