On Salem set

Let $\mu$ be a Borel probability measure on $R^d$. We say that $\mu$ is an $M_\beta$-measure if its Fourier transformation $\widehat{\mu}$ possesses the following property:
$$\widehat{\mu}(\xi)=o(|\xi|^{-\beta}), |\xi|\to \infty.$$

We define the Fourier dimension of $\mu$ as 

$$\dim_F\mu=\sup\{\alpha\in[0,d]:\text{$\mu$ is an  $M_{\alpha/2}$-measure }\}.$$

Then it is easy to verify that 

$$\dim_F\mu=\liminf\limits_{|\xi|\to \infty}\frac{-2\log |\widehat{\mu}(\xi)|}{\log |\xi|}^ d.$$

Define the Fourier dimension of a Borel set $E\subset R^d$ as 

$$\dim_F E=\sup_{\mu\in \mathcal{P}(M)}\dim_F\mu.$$

Fact 1: $\dim_F E\le \dim_H E.$

In fact, recall that , for $s\ge 0$ the $s$-potential at a point $x\in R^d$ is defined as 

$$\phi_s(x)=\int \frac{d\mu(y)}{|x-y|^s}.$$

It can be writtten as 

$$\phi_s(x)=(|x|^{-s} \ast \mu)(x)=\int |x-y|^{-s}d\mu(y).$$

On the other hand,  the Fourier transformation of $|x|^{-s}$ is $c |x|^{s-d}$ (see Some useful facts on Fourier transformation). So, we have

$$\widehat{\phi_s}(\xi)=c|\xi|^{s-d}\cdot \widehat{\mu}(\xi).$$

Then, by Parseval's theorem (again, see Some useful facts on Fourier transformation) we can obtain the following expression of $s$-energy:

$$I_s(\mu)=\int\phi_s(x)d\mu(x)=c(2\pi)^d\int |\xi|^{s-d}|\widehat{\mu}(\xi)|^2d\mu(\xi). $$

Now, suppose that $t<\dim_F E.$ Then there exists a measure $\mu\in \mathcal{P}(E)$ such that $|\widehat{\mu}(\xi)|\le b |\xi|^{-t/2}$.

Therefore, if $0<s<t$

$$I_s(\mu)\le c_1 \int_{|\xi|\le 1}|\xi|^{s-d}d\mu +c_2 \int_{|\xi|> 1}|\xi|^{s-d}|\xi|^{-t}d\mu<\infty, $$

which implies that $\dim_H E\ge t.$ So, $\dim_F E\le \dim_H E.$

The above inequality can be strict. For example, the Fourier dimension of ternary Cantor set is zero (we can show that the Cantor measure has no Fourier decay, i.e., $\widehat{\mu_C}(\xi)\nrightarrow 0$, see Salem-Bluham's Construction of Salem Sets by Xianghong Chen.)

It has been known that the Huasdorff dimension $\dim_H E$ describes a metric property of $E$, whereas the Fourier dimension measures an arithmetic property of $E.$ As a further example of this aspect, we mention that every set $E\subset R^d$ with positive Fourier dimension generates $R^d$ as a group, see Geometry of Sets and Measures in Euclidean Spaces by P. Mattila or Some Random Series of Functions by Kahane, J.-P.. 

We say that a Borel set $E$ is Salem set if $\dim_F E= \dim_H E.$

See, Sectorial local non-determinism and the geometry of the Brownian sheet by Khoshnevisan, Wu and Xiao, or Falconer,  Chapter 4 in Fractal geometry.

Fact 2: Given $0<s<t< 1$, there exists a Borel set $E$ such that $\dim_F E=s$ and $\dim_H E=t.$

See, T. W. Korner, Hausdorff and Fourier dimension, Studia mathematica, 206 (2011), or Christian Bluhm, On a theorem of Kaufman: Cantor-type construction of linear fractal Salem set, Ark. Mat., 36 (1998).