• Quasi-random numbers (almost)

    From Arthur Rubin@21:1/5 to All on Wed Oct 11 19:06:08 2017
    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

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