Let $\epsilon_n$ be a sequence in $\{-1,1\}^{\mathbb Z_+}$. For simplicity, assume that $\epsilon_n$ is just the Thue-Morse sequence with symbols $1$ and $-1$ (although the following definition is supposed to make sense for more general $\epsilon_n$ too)

We define partial Fourier amplitudes

$$G_N(q)=\sum_{1\leq n\leq N} \epsilon_n \exp(inq)$$

and the corresponding intensities


Does the limit

$$d\mu(q)=\lim_{N\to\infty} \left[ S_N(q) \frac{dq}{2\pi}\right]$$

make sense? How do we prove rigorously that it exists?

For context, this definition of $d\mu(q)$ (the Fourier intensity measure) comes from (2.3) in the following paper:

Luck, J. M. "Cantor spectra and scaling of gap widths in deterministic aperiodic systems." Physical Review B 39, no. 9 (1989): 5834.

This paper can be found here (paywalled)

I asked this question previously on Math Stackexchange here. But I think it might be a better fit for Math Overflow


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