COINCIDENCES IN POISSON PATTERNS 1023 



is always of volume ^ jS{8). Hence 



pj^, ^ pj[l - jSi8)/V], 

 or 



fc-i 



P^ ^ n [1 - jS(8)/V]. (4-3) 



3 = 1 



When {k — l)S(8) > T" the above argument fails because the later 

 terms of the product are negative; in this case we use the trivial bound 

 Ph ^ 0. Combining (4-2) with. (4-3) the lemma follows. 



Once more the bound may be expected to be almost correct if \'VS{28) 

 is small and if most of the region T^ lies farther than 8 away from its 

 boundary. The bound is also correct for non-spherical neighborhoods 

 (see discussion of previous theorem). 



When V/S{8) is large, the sum (4-1) is unwield}'. If we let H equal 

 V/S{8), we may rewrite the typical term in the sum as 



^^lia-j/H) =^^^i/(//-i) ... (//-A- + 1). 



If H happens to be an integer, this equals 



(f ) (mnr, 



so that the complete sum (4-1) eci[ua]s 



e-"' (l - ^y. (4-4) 



We will now prove that if ^ is not an integer, the sum always exceeds 

 (4-4), so that (4-4) is a lower bound in all cases. We ^^ish to prove that 



[H] + l k 



1 + E r, ^(^ - 1) • • • (// - A- + 1) ^ (l-f .r)^ (4-5) 



fc=l Kl 



for any positive H, in which event the theorem follows with 



X = ^ and H = T7^(5). 



The inequahty (4-5) will be proved by induction on [H]. If [H] = 0, 

 then we are required to show that 



1 + Hx ^ (1 + •r)'' 



