By Carlos A Berenstein

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Applying the ~ > O, ]~(t+in+u) l ! T~lnle Alnl M~(t) max lu I<_1 Therefore, by (67) and (68), (69) f(~-t) ^ +"in) - ~ ( t ) [ a t Ig(t I S ~(~-t)~(t)dt > C2eX(g) _ T l l q l e A ql ; e x p [ x ( ~ - t ) Using again the superadditivity X(g-t) - ~(t) < X(~) - log(l+t2) • lql ! 6 y > for some 6 < e, - 9(t)]dt of X (cf. (55)), we get Then, for lq I sufficiently small, say (69) yields (70) l cz >- T eX(g) Now, let us define m' as (71) m'(E) = C3exp[6AInl - ~ t l ( ~ ) ] ; C 3 = 2CCIC2 le3As ; 57 and, for fixed, we set z ° = Xo+iY o where ~,~ are as above (see (60)).

J (13) If the sequence {H s} is defined (14) then by by by the p(Hj) (13), (I0), the sequence Hsz rz = conditions j (j c z) , Hj. -" ~ , H j /j ÷ ~, { H i } j > 1 is concave, and Set (15) ~ and choose ~. ~s p o s i t i v e = I + aI so that o@ (16) < ~ Ss C T s=l For any for some (17) Suppose integers ~ first ~ ~ in Cn we write IOl k(~) Indeed, < aZ+l and aq We claim < as s~ = s=l _< tet that Cexp(-iw(~) if k is w r i t t e n k then Then, ~ and q, @ ~ 0. (18) = ~ + in = ~ + i@~(~). e . , exp(-Xm(~) + r z l n [) < s=l For the estimate of the second sum, the geometric properties of the function p(Ixl) have to be used.

Vi) is easy m E Af(a~) >_ const. > 0. k(~) Thus, choosing = C- 1 Then (52) the m ~ ~f(@) B C A(m;~) and k as above. by We claim (6) and + 1, for some that (50), ~ ~ s e X p [ ( H s - A ) t~[ - ( s + ~ ) ~ ( ~ ) + ~ ( ~ ) k ( w ( ~ ) ) ] s=l for HSo> I~I _> E, A, and family from so large : I¢1 < E < m* s ~(£) const, m*(~) such that (51), ~A~(~) is a B A U - s t r u c t u r e , a function m(~) E > 0 we obtain > C-I~s mine o for each A = Rs to check. e. 0 Let us choose • m = m({Cn};A;... ). Now let (51) To prove 2 Rs }, o condition k(~)/m(~) Cn = {x : [xl 0 > it suffices that to find If m is given by (49) and {Vj} as above, sequence of points Igjl : ~ and gj,{j s aVj for ~(gj) = Cj+ I- Cj.