A Gabor system is a set of time-frequency shifts S(g, Lambda) = {e(2 pi ibx
) g(x - a)}((a,b)is an element of Lambda) of a function g is an element of
L-2(R-d). We prove that if a finite union of Gabor systems boolean (ORk=1S)
-S-r(g(k), Lambda(k)) forms a frame for L-2(R-d) then the lower and upper B
eurling densities of Lambda = boolean ORk=1r Lambda(k) satisfy D- (Lambda)
greater than or equal to 1 and D+(Lambda) < infinity. This extends recent w
ork of Ramanathan and Steger. Additionally, we prove the conjecture that no
collection boolean ORk=1r {g(k)(x - a)}(a is an element of Gamma k) of pur
e translates can form a frame for L-2 (R-d). (C) 1999 Academic Press.