  Recently the so-called Atiyah conjecture about l^2-Betti numbers has been
disproved. The counterexamples were found using a specific method of computing
the spectral measure of a matrix over a complex group ring. We show that in
many situations the same method allows to compute homology gradients, i.e.
generalizations of l^2-Betti numbers to fields of arbitrary characteristic. As
an application we point out that (i) the homology gradient over any field of
characteristic different than 2 can be an irrational number, and (ii) there
exists a finite CW-complex with the property that the homology gradients of its
universal cover taken over different fields have infinitely many different
values.
