Abstract <p>A new class of rationally infinitely divisible probability distributions is an essential extension of the fundamental class of infinitely divisible laws. By definition, a distribution function is called rationally infinitely divisible if its convolution with some infinitely divisible d.f. is infinitely divisible. The characteristic functions of such distribution functions admit the Lévy–Khintchine representation where the spectral function has a bounded total variation on the real line and it can be non-monotonic. Now this new class is investigated and it gets various applications. There have been some papers published by now on criteria for the membership in this class. The conditions of all results are expressed in fact in terms of characteristic functions. In this paper, considering only discrete distributions at arbitrary points on the real line, we obtain conditions for the rationally infinite divisibility of their distribution functions expressed in terms of these points and their probabilities. We show that the membership in this class is significantly affected by how many elements among these points are linearly independent over the field of rational numbers.</p>