Distance requirements in antenna arrays?
can antenna arrays be decreased in size if the antenna elements themselves can be decreased in size?


No, they cannot. The antenna arrays work by constructive and destructive interference where, in one direction the radiation from antennas of the array is in the same phase, leading to a maximum in radiation, and in one direction the phases are opposite, leading to a null in the radiation. The interference pattern is a function of the phase difference of these different radiated fields and if the antennas would be in the same locations, there would be no interference.

The analytical way of analyzing an antenna array is called array factor. For N identical antennas, the array factor is:

$$ AF = \Sigma_{i=1}^N w_i e^{-j k r_i} $$

where \$w_i\$ is the weight of the specific antenna, \$ r_i\$ is the vector defining the position of the antenna, and \$k\$ is the wave vector.

If you just want to get a good gut feeling on how the locations and phase differences affect, have a look at the Python code I used to answer another SE question: https://nbviewer.jupyter.org/gist/Diimu/71010f1ffee95e6fe0082ad1dffda5c3/analytical_dipoles.ipynb . The tool uses ideal Hertz's dipoles and neglects their mutual coupling, but can be used to model surprisingly complicated things.

If you don't have Python installed, you can try it out in e.g., https://jupyter.org/try . The python is a quite easy language to play with if you have any programming experience. Below you can see the directivity pattern of an array of two horizontal electric dipole antennas that are fed with the same phase and amplitude. The distance is varied from 0 to 1 wavelength in a loop. As you can see, the case where the distance is \$0.1\lambda\$ the radiation pattern is virtually identical with \$0\lambda\$. At \$0.5\lambda\$ you see a high directivity as the two dipoles cancel each other out in the direction of the displacement and amplify each other \$90^\circ\$ away from that. 

For further reading see below, or pretty much any antenna theory book:


http://www.antenna-theory.com/arrays/arrayfactor.php 
http://www.faculty.jacobs-university.de/jwallace/xwallace/courses/ap/ch5.pdf