Deviations of ergodic sums for toral translations II. Boxes
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ABSTRACT We study the Kronecker sequence {nα}n≤N on the torus Td when α is uniformly distributed on Td . We show that the discrepancy of the number of visits of this sequence to a random box, normalized by lnd N, converges as N → ∞ to a Cauchy distribution. The key ingredient of the proof is a Poisson limit theorem for the Cartan action on the space of d + 1 dimensional lattices.
CONTENTS 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 3. Negligible contribution of non-resonant terms . . . . . . 4. Poisson distribution of small divisors . . . . . . . . . . . . 5. Reduction to dynamics on the space of lattices . . . . . . 6. Poisson limit theorem for almost independent rare events 7. Rate of equi-distribution of unipotent flows . . . . . . . . 8. Poisson limit theorem for the diagonal action . . . . . . . 9. Small boxes . . . . . . . . . . . . . . . . . . . . . . . . . 10.Continuous time . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . Publisher’s Note . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Norms . . . . . . . . . . . . . . . . . . . . . . Appendix B: Independence . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction 1.1. Equidistribution of Kronecker sequences on Td . — It is well known that the orbits of a non resonant translation on the torus Td = Rd /Zd are uniformly distributed. A quantitative measure of uniform distribution is given by the discrepancy function: for a set C ⊂ Td let D(α, x, C , N) =
N−1
1C (x + nα) − Nν(C )
n=0
where (α, x) ∈ Td × Td , 1C is the characteristic function of the set C and ν is the Haar measure on the torus. (We will sometimes write νd if we want to emphasize the dimension of the torus). Uniform distribution of the sequence x + nα on Td is equivalent to the fact that, for regular sets C , D(α, x, C , N)/N → 0 as N → ∞. A step further is the study of the rate of convergence to 0 of D(α, x, C , N)/N. Already with d = 1, it is clear that if α ∈
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