By Jean Bertoin

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4. Fix a real number a. } to (a, a + 27r) N (ordered). (2) For each integer k > 1, denote by Wk the locally closed set in PN,s' consisting of those f which have a zero at T = eia of exact multiplicity k. This bijection is continuous on Wk. PROOF. It suffices to treat the case a = 0. (1) Any element f in PN,St-{1} lies in PN,Z[E,2a-E] for some c > 0. If f --+ f is a Cauchy sequence in PN,SI, we claim that each fn, itself lies in PN,Z[E/2,2a-E/2] as soon as n >> 0. To see this, take any point a in Z[-e/2,,-/2]- Then we have dist(a, Z[e, 27r - E]) > e/7r (the circular distance is > e/2), and hence our f in PN,Z[e,2a-E] satisfies If (a) I ?

Tr, E Clump(a)(s)Ta = a>O Sep(b)(s)T6 = b>O Sep(b)(s)(1 + T)b, b>O C1ump(a)(s)(T - 1)a. a>O PROOF. The first identity is, coefficient by coefficient, the identity of the lemma. The second identity is obtained from the first by the change of variable T - T-1. QED 2. 11. For each a > 0 in Z', we have the identity Sep(a)(s) _ (-1)"-LBinom(n, a) Clump(n)(s). 12. For each a > 0 in Zr, and each integer m > E(a), we have the inequalities (-1)n-a Binom(n, a) Clump(n)(s) < Sep(a)(s) a

A, H(N), offsets c), p,(univ, offsets c))d Haar JH(N) discrep(µ(A, G(N), offsets c), p(univ, offsets c))dHaar. G(N) 1. STATEMENTS OF THE MAIN RESULTS 28 In case d), we have f discrep(u(A, H(N), offsets c), u(univ, offsets c))d Haar H(N) = either discrep(g(A, SO(2N), offsets c), u(univ, offsets c))dHaar fSO(2N) or f discrep(p,(A, O(2N), offsets c), u(univ, offsets c))dHaar, (2N) depending on whether or not H(N) acts by inner automorphism on SO(2N). 6 extended). 6 holds. Let N be an integer > 1, G(N) C H(N) compact groups in one of the following four cases: 'ti a) G(N) = SU(N) C H(N) C normalizer of G(N) in U(N), b) G(N) = SO(2N + 1) C H(N) c normalizer of G(N) in U(2N + 1), c) G(N) = USp(2N) C H(N) C normalizer of G(N) in U(2N), d) G(N) = SO(2N) C H(N) C normalizer of G(N) in U(2N).