New York Journal of Mathematics
Volume 20 (2014) 195-208


David Ralston

Generic (1/2)-discrepancy of {n θ + x}

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Published: February 19, 2014
Keywords: discrepancy, irrational rotation, renormalization, substitution
Subject: Primary: 11K38, Secondary: 37E20, 37B10

We study the pointwise rate of divergence of the sequence of discrepancy sums
Si(x) = ∑j=0i-1[0,1/2)[1/2,1))(x+j θ)
for (Lebesgue) generic rotation parameter θ. Almost-sure upper and lower bounds for the rate of divergence are given by a Khinchin-like criterion related to convergence of a certain integral. Concluding remarks address the impossibility of finding a generic asymptotic rate of divergence as well as the related study of how frequently Si(x)=0.

Author information

Depatment of Mathematics/CIS, SUNY College at Old Westbury, PO 210, Old Westbury, NY 11568