Lattice point counting in sectors of Hyperbolic 3-space

• Let G be a cocompact discrete subgroup of PSL2(.) and denote by. the three-dimensional upper half-space. For a p I., we count the number of points in the orbit Gp, according to their distance, arccosh X, from a totally geodesic hyperplane. The main term in n dimensions was obtained by Herrmann for any subset of a totally geodesic submanifold. We prove a pointwise error term ofO(X3 2) by extending the method of Huber and Chatzakos-Petridis to three dimensions. By applying Chamizo's large sieve inequalities, we obtain the conjectured error term O(X1+ e) on an average in the spatial aspect. We prove a corresponding large sieve inequality for the radial average and explain why it only improves on the pointwise bound by 1/ 6.

Ämnesord

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

Nyckelord

Riemann Surfaces

Large Sieve

Plane

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