I have attached a paper about Planar Interferometer Arrays. Now I’m trying to understand this paper but something confuses me. I am very thankful if somebody can help me with it.

1:

It seems to me that direction cosines and phase differences are 1:1 correspondent, that is, a phase difference can lead to a fixed direction cosine. Is it correct?

2:

What is meant by „unmeasured integer portions of the relative phase measurements between all antennas pairs“? Does it have something to do with resolving ambiguities?

The phase difference between two antennas specifies the cosine of the angle of arrival. But with only one pair of antennas the source is only known to be on the surface of a cone with it's axis through the two antennas. With sufficient antenna pairs the direction of intersection of many cones can be computed. All cones form circles on the spherical map of all possible DOA.

Since phase differences are modulo 360°, when the dimensions of the array exceed one wavelength, there are ambiguous phase solutions. It is necessary to identify the integer number of cycles that are missing and apply them when computing the cosines of the angles.

Your observation 1: “that a phase difference can lead to a fixed direction cosine”, is correct only if the ambiguity is not present, probably because the array is small.

Your observation 2: “unmeasured integer portions of the relative phase measurements between all antennas pairs”, is referring to the missing complete cycles.

That work was in 1983. Since then, computers have gained sufficient speed to predict the phases expected from every point on a one degree grid in space, then search the DOA space for the least squares best match to the current measured differential phases. There is no ambiguity because the ambiguous results are suppressed by the least squares score. In 1989 I was doing that on an array of 18 antennas, with an Intel 386. Things have changed.

That is because radio engineers like to measure things in wavelengths, not radians.

The baselines between the elements of the array are being measured in wavelengths.
That is shown a few lines above the 0 ≤ øi < 1, where the d. vector is defined in wavelengths.