In few papers (see, for example, here, the bottom of the left column on the page 6, or here, the upper part of the page 5) I've met the strange calculations using the **constant gauge field**

$$

A_{\mu}(x) = (0,0,0,A_{3} = \text{const}), \quad\text{or}\quad A_{\mu}(x) = (A_{0} = \text{const},0,0,0)

$$

By using these fields authors obtain observable effects like chiral and vector currents.

One might think that these constant gauge fields can be gauged away, but the authors of the first linked article say that (at least about constant $A_{3}$)

One might think that a constant gauge field could be gauged away, but this is not possible by a gauge transformation satisfying the periodic boundary condition.

I don't understand this statement. Could You clarify it? Also I don't understand what is the problem with $A_{0} = \text{const}$.