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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