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### Open Access Journals

JMD

A closed geodesic on the modular surface gives rise to a knot on the
3-sphere with a trefoil knot removed, and one can compute the linking
number of such a knot with the trefoil knot. We show that, when
ordered by their length, the set of closed geodesics having a
prescribed linking number become equidistributed on average with
respect to the Liouville measure. We show this by using the
thermodynamic formalism to prove an equidistribution result for a
corresponding set of quadratic irrationals on the unit interval.

JMD

For manifolds with geodesic ﬂow that is ergodic on the unit tangent bundle, the Quantum Ergodicity Theorem implies that almost all Laplacian eigenfunctions become equidistributed as the eigenvalue goes to inﬁnity. For a locally symmetric space with a universal cover that is a product of several upper half-planes, the geodesic ﬂow has constants of motion so it cannot be ergodic. It is, however, ergodic when restricted to the submanifolds deﬁned by these constants. Accordingly, we show that almost all eigenfunctions become equidistributed on these submanifolds.

JMD

We give upper and lower bounds for Diophantine exponents measuring how well a point in the plane can be approximated by points in the orbit of a lattice $\Gamma < {\rm{S}}{{\rm{L}}_2}\left( {\mathbb{R}} \right)$ acting linearly on ${\mathbb{R}^2}$. Our method gives bounds that are uniform for almost all orbits.

JMD

We prove a quantitative version of Oppenheim's conjecture for generic ternary indefinite quadratic forms. Our results are inspired by and analogous to recent results for diagonal quadratic forms due to Bourgain [

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