(**Preliminaries:**) 1.) Let $S\subset\mathbb{R}^n$ and define $\mathcal{M}(S) = \{\text{$\mu$ a Borel measure}: \text{$0 < \mu(S) < \infty$ and $\mathrm{support}(\mu)\subset S$}\}$.

2.) Define the Fourier transform of a measure as $\hat{\mu}(x):= \int_{\mathbb{R}^n}e^{-2\pi i\xi \cdot x}d\mu(\xi)$.

3.) Define the Fourier dimension of a set $S\subset\mathbb{R}^n$ as

$$\mathrm{dim}_FS := \sup\{s\in \left[0,n\right]: \exists \mu\in \mathcal{M}(S):\forall x\in\mathbb{R}^n:\left|\hat{\mu}(x)\right|\leq \left|x\right|^{-s/2}\}$$

(**Remark:**) I was a bit hesitant to post this question here since there is a non-zero chance for it being completely trivial, but as I (honest to God) could not find any discussion on this in books such as Mattila's *Fourier Analysis and Hausdorff Dimension* or *Geometry of Sets and Measures in Euclidean Spaces* or in exercise sections of such relevant books, here I am.

(**Question:**) Given $-\infty<a<b<\infty$, what is the Fourier dimension of the interval $\left[a, b\right]\subset\mathbb{R}$ and what measure $\mu\in \mathcal{M}\left(\left[a, b\right]\right)$ gives it?

My naïve first thought was to just use the one dimensional Lebesgue measure restricted to $\left[a, b\right]$. However, then for $x\neq 0$ we get

$$|\hat{\mu}(x)|^2 = \left|\frac{i}{2\pi x}\left(e^{-2\pi i xb} - e^{-2\pi ixa}\right)\right|^2 = \frac{1 - \cos\left(2\pi x(b - a)\right)}{2\pi^2x^2}$$

whence $|\hat{\mu}(x)|^2 \leq \left|x\right|^{-s}\Longleftrightarrow 1 - \cos\left(2\pi x(b - a)\right)\leq 2\pi^2\left|x\right|^{2-s}, s\in \left[0,1\right]$

By taking $e.g. b = 4, a = -2, x = 0.05$ we see that if $\left[a, b\right]$ were to have a Fourier dimension equal to one, then $\mu$ will not give it as the LHS is equal to $\approx 1.309$ while the RHS is equal to $\approx 0.987$.

Any ideas how the measure $\mu$ should be constructed? Also, do you happen to know a good source which works through a bit more elementary examples (like this) of the Fourier dimension?