By Diamond H. (ed.)
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Additional info for Analytic Number Theory
One first proves a generalization of Theorem 3 in which xn is replaced by a tuple xn ; : : : ; xnCm 1 of variables. x 0 ; xn ; : : : ; xnCm 1 / looking the same as in Theorem 3 except that the subscript j becomes a multi-index ˛ 2 Nm with j˛j Ä d . And now the point is that in successively eliminating the quantifiers 9xnCm 1 ; 9xnCm 2 ; : : : ; 9xn (as above) we apply the D function to terms involving the variables x 0 only. x /, we might not have guaranteed the regularity of F with respect to any of the variables xn ; : : : ; xnCm 1 .
A; p1 ; ::; pk /j. Fix > 0. p/ N > g/ ! 0 as k ! 1. 7. MN =M0 /invariant for some small model M0 . 8 (NIP). MN =M0 /-invariant). M0 / W . M0 /. 8 we can form “nonforking amalgams” of global Keisler measures (generalizing the case p ˝ q for types mentioned earlier). y/ any global Keisler measure. x; y/ over M . x; y/ over MN . MN =M0 /-invariant. We define Z . y//. x; y// D . M0 / Some explanations of the formula above are in order. We write . x; q// because by M0 -invariance, for b 2 MN , . b=M0 /.
Let C be a family of “events” which has finite VC-dimension. A; p1 ; ::; pk /j. Fix > 0. p/ N > g/ ! 0 as k ! 1. 7. MN =M0 /invariant for some small model M0 . 8 (NIP). MN =M0 /-invariant). M0 / W . M0 /. 8 we can form “nonforking amalgams” of global Keisler measures (generalizing the case p ˝ q for types mentioned earlier). y/ any global Keisler measure. x; y/ over M . x; y/ over MN . MN =M0 /-invariant. We define Z . y//. x; y// D . M0 / Some explanations of the formula above are in order. We write .
Analytic Number Theory by Diamond H. (ed.)