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3.2Useful030109 - 1 3.2 Some Useful Lemmas DEFINITION 3.2.1...

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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.
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2 Let A N be nonempty, where fA 2N A. Let f(min(A),min(A),min(A)) = 2 k 2. Then 2 k fA 2N. Therefore 2 k A. Suppose j k and 2 j A. Then f(2 j ,2 j ,2 j ) = 2 j+1 fA. We have thus established by induction that for all j k, 2 j A.
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