For a stable subordinator Yt of index [alpha], 0<[alpha]<1, the occupation measure[mu](A)={t[set membership, variant][0,1] : Yt[set membership, variant]A}is known to have (with probability 1) the property thatIn order to obtain an interesting spectrum for the large values of [mu](x-r,x+r), we consider the setwhere c[alpha] is a suitable constant. It is shown that B[theta]=[empty set][combining character] for [theta]>1, and B[theta] [not equal to] [empty set][combining character] for 0[less-than-or-equals, slant][theta][less-than-or-equals, slant]1; moreover, dim B[theta]=Dim B[theta]=[alpha](1-[theta]1/(1-[alpha])).