Spatial sampling frequency equation
The spatial sampling frequency of a linear array with equidistant sensor (antenna) spacing on the $x$-axis is simply given by

$$f_x=\frac{1}{d}\tag{1}$$

where $d$ is the distance between the sensors (cf. Eq. $(35)$ in the document you linked to). What is meant by the equation in your question is the spatial frequency of the incoming wave along the $x$-axis, according to the $x$-component of the direction vector $\mathbf{\alpha}$ (cf. Eq. $(13)$ in the document).

To avoid spatial aliasing you need to guarantee

$$\frac{1}{d}&gt;2\mathbf{\alpha}_x=2\frac{\sin\theta\cos\phi}{\lambda}\tag{2}$$

for all possible angles $\theta$ and $\phi$. The maximum value of the numerator of the fraction on the right-hand side of $(2)$ is $1$, and the minimum value of the denominator is the minimum possible wavelength $\lambda_{\text{min}}$:

$$\frac{\sin\theta\cos\phi}{\lambda}\le \frac{1}{\lambda_{\text{min}}}\tag{3}$$

Combining $(2)$ and $(3)$ gives

$$d&lt;\frac{\lambda_{\text{min}}}{2}\tag{4}$$

which corresponds to Eq. $(38)$ in the document.