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For a sequence (x_n) of reals that are equidistributed modulo one (such as (prime-) multiples of irrationals, or most geometric sequences) does the (properly scaled) sum of the remainders converge to 1/2 in some sort of central limit theorem sense?
What is this field of study called and what are good keywords to look for?
For equidistributed sequences, the Hlawka-Kocsma Inequality show that convergence occurs. It not a CLT type convergence. If the sequence is deterministic (a sequence of the fractional parts of square roots of primes), then the average of the sum will converge to 1/2 (actual convergence). The error (accuracy or precision may be a better word as every number is exact) is given by e
If the sequence is randomly chosen (begging the question of what this means), the discrepancy is proportional to 1/Sqrt(N) for N terms and so this mimics the precision of the statistical estimates.
Check out Quasi Monte Carlo, Kocsma-Hlawka Inequality, Low Discrepancy Sequence, and a bunch more.
