The sequence {nα} mod 1 fills [0,1) uniformly when α is irrational. See how rational vs irrational α behaves.
Weyl's Theorem (1916): For irrational α, the sequence nα mod 1 is equidistributed mod 1: for any interval [a,b]⊂[0,1), exactly the fraction (b−a) of the first N terms land there as N→∞. Proved via exponential sums: |Σe^{2πiknα}|/N → 0 for k≠0. Rational α gives periodic orbits that miss most of [0,1).
Timeline: each point = {nα} on [0,1)
Circle (angle = 2π·{nα})
Discrepancy: —Gap ratio: —
Histogram of {nα} in 32 bins (should be flat for irrational α)