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Effective equidistribution of shears and applications

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Abstract

A “shear” is a unipotent translate of a cuspidal geodesic ray in the quotient of the hyperbolic plane by a non-uniform discrete subgroup of \({\text {PSL}}(2,\mathbb {R})\), possibly of infinite co-volume. We prove the regularized equidistribution of shears under large translates with effective (that is, power saving) rates. We also give applications to weighted second moments of GL(2) automorphic L-functions, and to counting lattice points on locally affine symmetric spaces.

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Notes

  1. By “spectral gap” we always mean the distance between the first eigenvalue \(\lambda _{1}\) and the base eigenvalue \(\lambda _{0}\) of the hyperbolic Laplacian; see Sect. 2.5.

  2. Of course one can instead order by wordlength in \(\Gamma \), as is done in [3], to restore equidistribution and apply the Affine Sieve.

  3. Equivalently, non-elementary, that is, not virtually abelian.

  4. For surface groups, being geometrically finite is equivalent to being finitely generated.

  5. Roughly speaking, the critical exponent measures the asymptotic growth rate of \(\Gamma \); see Sect. 2.1.

  6. Added in print: In private communication, Hee Oh has notified us that she and Nimish Shah have an unpublished manuscript in which they obtain a result similar to (1.16) in the lattice case.

  7. Recall that the limit set, \(\Lambda \), of \(\Gamma \) decomposes disjointly into cusps (i.e., parabolic fixed points) and radial limit points (also called “points of approximation”); the complement \(\partial \mathbb {H}{\setminus }\Lambda \) is called the free boundary (which is empty if \(\Gamma \) is a lattice). See §2.1.

  8. We should note that we are using a non-standard definition for the Eisenstein series, where we multiply the series by \(\tfrac{1}{\omega }\) instead of by \(\tfrac{1}{\omega ^s}\). Using the standard definition instead will result in minor changes to the regularized Eisenstein series in the lattice case.

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Acknowledgements

The authors would like to express their gratitude to Jens Marklof, Curt McMullen, and Peter Sarnak for many enlightening comments and suggestions. The second author would especially like to thank Valentin Blomer, Farrell Brumley, and Nicolas Templier for many hours of discussion about this problem; see also the related work in [1].

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Authors and Affiliations

  1. Boston College, Boston, MA, USA

    Dubi Kelmer

  2. Rutgers University, New Brunswick, NJ, USA

    Alex Kontorovich

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  1. Dubi Kelmer
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Correspondence to Alex Kontorovich.

Additional information

Communicated by A. Venkatesh.

Kelmer is partially supported by NSF Grant DMS-1237412 and NSF Grant DMS-1401747.

Kontorovich is partially supported by an NSF CAREER Grant DMS-1254788/DMS-1455705, an NSF FRG Grant DMS-1463940, and an Alfred P. Sloan Research Fellowship.

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Kelmer, D., Kontorovich, A. Effective equidistribution of shears and applications. Math. Ann. 370, 381–421 (2018). https://doi.org/10.1007/s00208-017-1580-9

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