Ergodic Theoretic Proof of Equidistribution of Hecke Points

The Distribution of Closed Geodesics on the Modular Surface, and Duke’s Theorem

We give an ergodic theoretic proof of a theorem of Duke about equidistribution of closed geodesics on the modular surface. The proof is closely related to the work of Yu. Linnik and B. Skubenko, who in particular proved this equidistribution under an additional congruence assumption on the discriminant. We give a more conceptual treatment using entropy theory, and show how to use positivity of ...

Ergodic Theoretic Characterization of Left Amenable Lau Algebras

Small Solutions to Linear Congruences and Hecke Equidistribution

The purpose of the present note is to give an explicit application of equidistribution of Hecke points to the problem of small solutions of linear congruences modulo primes. Let p be an odd prime. For a random system of d linear congruences in n variables modulo p, we shall ask how many “small” solutions it has, i.e. solutions in Z of size about pd/n, the natural limiting point. We will show th...

Equidistribution, Ergodicity and Irreducibility in CAT(-1) spaces

We prove an equidistribution theorem à la Bader-Muchnik ([3]) for operator-valued measures associated with boundary representations in the context of discrete groups of isometries of CAT(-1) spaces thanks to an equidistribution theorem of T. Roblin ([22]). This result can be viewed as a von Neumann’s mean ergodic theorem for quasi-invariant measures. In particular, this approach gives a dynamic...

Equidistribution of Expanding Translates of Curves and Dirichlet’s Theorem on Diophantine Approximation

We show that for almost all points on any analytic curve on R which is not contained in a proper affine subspace, the Dirichlet’s theorem on simultaneous approximation, as well as its dual result for simultaneous approximation of linear forms, cannot be improved. The result is obtained by proving asymptotic equidistribution of evolution of a curve on a strongly unstable leaf under certain parti...