We obtain large gap asymptotics for Airy kernel Fredholm determinants with any number m of discontinuities. These m-point determinants are generating functions for the Airy point process and encode probabilistic information about eigenvalues near soft edges in random matrix ensembles. Our main result is that the m-point determinants can be expressed asymptotically as the product of m 1-point determinants, multiplied by an explicit constant pre-factor which can be interpreted in terms of the covariance of the counting function of the process.1 24 e ζ ′ (−1) of the multiplicative constant. For m = 1 with s > 0, it is notationally convenient to write s = e −2πiβ with β ∈ iR, and it was proved only recently by Bothner and Buckingham [13] thatwhere G is Barnes' G-function, confirming a conjecture from [8]. The error term in (1.5) is uniform for β in compact subsets of the imaginary line.We generalize these asymptotics to general values of m, for s 2 , . . . , s m ∈ (0, 1], and s 1 ∈ [0, 1], and show that they exhibit an elegant multiplicative structure. To see this, we need to make a change of variables s → β, by defining β j ∈ iR as follows. If s 1 > 0, we define β = (β 1 , . . . , β m ) bys m for j = m, (1.6) and if s 1 = 0, we define β 0 = (β 2 , . . . , β m ) with β 2 , . . . , β m again defined by (1.6). We then denote, if s 1 > 0, E( x; β) := E m j=1 e −2πiβjN (x j ,+∞)