1
3.2. Some Useful Lemmas.
DEFINITION 3.2.1. The standard pairing function on N is the
function P:N
2
→
N due (essentially) to Cantor:
P(n,m) = (n
2
+m
2
+2nm+n+3m)/2
≥
n,m.
It is well known that P is a bijection, and also that for
all n
≥
0, [0,n(n+1)/2)
⊆
P[[0,n)
2
]. In addition, P is
strictly increasing in each argument.
Let T:N
2
→
N be such that T(2n,2m) = P(n,m), T(2n,2m+1) =
T(2n+1,2m) = T(2n+1,2m+1) = 2n+2m+2. Then for all n
≥
0,
[0,n(n+1)/2)
⊆
T[([0,2n)
∩
2N)
2
]. Hence for all n
≥
8,
every element of [0,n
2
/8) is realized as a value of T at
even pairs from [0,n).
It is clear that T(2n,2m)
≥
(n
2
+2n)/2,(m
2
+2m)/2
≥
2n,2m.
Hence for n,m
≥
2, T(n,m)
≥
n,m.
LEMMA 3.2.1. There exists 3-ary f
∈
ELG
∩
SD such that the
following holds. Let A
⊆
N be nonempty, where fA
∩
2N
⊆
A.
Then fA is cofinite. We can also require that for all n
≥
0,
f(n,n,n)
∈
2N.
Proof: We define f
∈
ELG
∩
SD as follows. Let p,q
∈
[2
n
,2
n+1
), n
≥
0. Define f(2
n
,p,q) = min(2
n+1
+T(p-2
n
,q-
2
n
),2
n+2
). Note that for n
≥
8, as p,q vary over the even
elements of [2
n
,2
n+1
), every value in [2
n+1
,2
n+2
) is realized.
Also note that for all n
≥
0, f(2
n
,2
n
,2
n
) = 2
n+1
.
For all n > 0, define f(n,n,n) to be the least 2
k
≥
2n;
f(0,0,0) = 2.
For all n < m < r, define f(r,n,n) = 2r+1, f(r,n,m) = 2r+2,
f(r,n,r) = 2r+3, f(r,m,n) = 2r+4, f(r,r,n) = 2r+5. For all
triples a,b,c, if f(a,b,c) has not yet been defined, define
f(a,b,c) = 2|a,b,c|+1.
It is obvious that f
∈
SD. To see that f
∈
ELG, we need
only examine the definition of f(2
n
,p,q), p,q
∈
[2
n
,2
n+1
),
where n is sufficiently large. If p,q
∈
[2
n
,2
n
+2
n-1
), then
obviously f(2
n
,p,q)
≥
2
n+1
≥
4|2
n
,p,q|/3. If p,q
∉
[2
n
,2
n
+2
n-
1
), then f(2
n
,p,q)
≥
2
n+1
+T(p-2
n
,q-2
n
)
≥
2
n+1
+ 2
n-1
≥
5p/4,5q/4. Also,f(2
n
,p,q)
≤
2
n+2
≤
2p,2q. Therefore f
∈
ELG.