Matthew Welsh

A Cauchy limit theorem for the Kronecker sequence

It is well known that for alpha and x in the d-dimensional torus with alpha not in any rational subspace, the Kronecker sequence {n alpha + x}, n = 0, 1, 2, ..., is equidistributed, and much work has been done to study the discrepancy of this sequence D(alpha, x, U) = # { n < N : n alpha + x in U} - N|U| for a subset U of the torus. Although there are many interesting variations, in this talk we focus on alpha and x independent, uniformly distributed random variables and U a rectangular box. In the 1960s, Kesten proved that for d = 1, the discrepancy has a Cauchy distribution with parameter rho log N with rho independent of U for irrational |U| (but may depend on |U| if it is rational). More recently, Fayad and Dolgopyat have extended this result for any d, however with a random perturbation of U by an affine linear transformation. In forthcoming joint work with Fayad and Dolgopyat, we are able to obtain the Cauchy limit (with parameter rho (log N)^d) without this additional perturbation, assuming that the side lengths of U are diophantine. The talk will outline our proof for d = 2 and will be roughly divided into three parts according to tools we use from Fourier analysis, probability theory, and homogeneous dynamics.