Student Co-author

HMC Undergraduate

Document Type

Article - preprint


Mathematics (HMC)

Publication Date



Given n vectors {i} ∈ [0, 1)d, consider a random walk on the d-dimensional torus d = ℝd/ℤd generated by these vectors by successive addition and subtraction. For certain sets of vectors, this walk converges to Haar (uniform) measure on the torus. We show that the discrepancy distance D(Q*k) between the kth step distribution of the walk and Haar measure is bounded below by D(Q*k) ≥ C1k−n/2, where C1 = C(n, d) is a constant. If the vectors are badly approximated by rationals (in a sense we will define), then D(Q*k) ≤ C2kn/2d for C2 = C(n, d, j) a constant.


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© 2004 Wiley Periodicals

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