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Oct 11, 2017, 9:06:11 PM10/11/17

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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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