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Let X be a non-negative random variable with EX < ∞, and let 1 ≤ n1 ≤ n2 ≤ n3 ≤ . . be a sequence of integers which is lacunary in the sense that nj+1 /nj > c for some c > 1 and all sufficiently large j. Then X nj converges almost surely to EX. 5. 16) we would see that almost surely the empirical means X n cannot deviate by more than a multiplicative error of 1+O(ε) from the mean EX. Setting ε := 1/m for m = 1, 2, 3, . . (and using the fact that a countable intersection of almost sure events remains almost sure) we obtain the full strong law.

Since all geodesics in the stadium hit the boundary, this in principle allows us to understand the distribution of an eigenfunction on the boundary in terms of the eigenfunction in the interior. Indeed, one can show that an eigenfunction which is uniformly distributed in phase space in the interior, will have a normal derivative which is uniformly distributed on the boundary (rigorous formulations of this fact date 48 1. Expository articles back to [GeLe1993]. Thus, by assumption, every eigenvector is uniformly distributed on the boundary.

2, so suppose inductively that s ≥ 2 and that the claim has already been proven for smaller s. We then look at the vertical torus Gs /(Γ ∩ Gs ) ≡ Td , where Gs is the last non-trivial group in the lower central series (and thus central). The quotient of the nilmanifold G/Γ by this torus action turns out to be a nilmanifold of one lower step (in which G is replaced by G/Gs ) and so the projection of the orbit (g n x)∞ n=1 is then equidistributed by induction hypothesis. 5. The strong law of large numbers 23 quotienting out by the diagonal action of the torus, is equidistributed with respect to some measure which is invariant under the residual torus Td × Td /(Td )∆ .

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