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Quasirandom numbers (almost)
Posted:
Oct 11, 2017 9:06 PM


With a irrational, let x(n) = {a n}, the fractional part of {a n}.
What can be said about (1) The "distribution" of sum (k = 1 to n) x(k). Any bounds (upper or lower) for the deviation of the sample CDF of x(k) from uniform, or the maximum or minimum gap. (2) With X(k) = {U + x(k)}, for a single U, uniform on (0, 1). What can be said about the distribution of S(n) = sum(k=1 to n)X(k). E(S(n)) = n/2, and I can calculate Var(S(n)) as an algebraic function of x(k), but I have not calculated asymptotic bounds.
 Arthur L. Rubin



