  In this paper we solve a problem posed by M.E. Rossi: {\it Is the Hilbert
function of a Gorenstein local ring of dimension one not decreasing? } More
precisely, for any integer $h>1$, $h \notin\{14+22k, \, 35+46k \ | \
k\in\mathbb{N} \}$, we construct infinitely many one-dimensional Gorenstein
local rings, included integral domains, reduced and non-reduced rings, whose
Hilbert function decreases at level $h$; moreover we prove that there are no
bounds to the decrease of the Hilbert function. The key tools are numerical
semigroup theory, especially some necessary conditions to obtain decreasing
Hilbert functions found by the first and the third author, and a construction
developed by V. Barucci, M. D'Anna and the second author, that gives a family
of quotients of the Rees algebra. Many examples are included.
