Proof of Theorem 3. 1.5 In this section we prove Proposition 3.1.4 and Theorem 3.1.5, the weaker version of our main result. We start by introducing some notation and classical results that we shall need. Recall that, for all non-negative integers n ≥ k, we define n k q k Y :=
Proof of Theorem 3. 2.5.1 Our proof relies on the backward induction. It is trivial that equation (3.43)- (3.46) hold for t = T as in chapter 3.1.4. Assuming equation (3.39) and equa- tions (3.43)-(3.46) hold for t ≥ k + ∆t, we now examine the case for t = k. Let u = (uk, u∗k+∆t, ..., u∗T ), from definition (3.37), (3.38) and the Tower Property, the utility function can be written as J(k, Xk, ψk; u) = E [Xu] — γ V ar [Xx] k,Xk,ψk T = Ek,X ,ψ hE 2 k,Xk,ψk T uk Xu∗ i k k k+∆t,Xk+∆t
Proof of Theorem 3. 2.5.2 From equation (3.2), we can determine the expected wealth and the second moment at time t + ∆t based on time t as Et,X ,ψ [Xu ] = Σ (1 + µt,i∆t)ψt,i ut,i (3.69) t t t+∆t Σ = αt,2ψt,2ut,2 + rt,1Xt — αt,Li ψt,Li (3.70) Li The variance of wealth at time t + ∆t in terms of those on time t is V art,X ,ψ [Xx ] = Σ σt,iσt,jρij ∆tψt,iut,iψt,jut,j (3.71) t t t+∆t i,j Σ t,2 i i =σ2 ∆t(ψt,2ut,2)2 — 2 Σ Dt,2L ψt,L ψt,2ut,2 Li + Dt,LiLj ψt,Liψt,Lj (3.72) Xx,Xx Thus we have
Proof of Theorem 3. 2.1 Roughly speaking, condition (2) says that the harmonic measures in Ω and Ω∗ are comparable in the sense that their ratio is bounded above and below. Condition
Proof of Theorem 3. 1.1 The total number of subsets of [n] having fewer than n1/4(log n)2 elements is 2o(n1/3). Therefore we can focus on B3-sets of size n1/4(log n)2 ≤ t < n1/3. In particular, by Theorem 3.2.1(i), |Zn| ≤ 2 o(n1/3) n1/3 Σ + t=n1/4(log n)2 cn t t3
Proof of Theorem 3. 2.1 The proof of Theorem 3.2.1 uses the following strategy. Suppose that a B3-set S ⊂ [n] of cardinality s is given and one would like to extend it to a larger B3-set. We will show that if S satisfies a boundedness condition (see Definition 3.3.9 below), then the number of such extensions is fairly small. Moreover, we also prove that almost all B3-sets of cardinality s are sufficiently bounded in the sense of Definition 3.3.9. Consequently, in order to provide an upper bound for the number Zn(t) of B3-sets of size t in [n], for some t > s, we
Proof of Theorem 3. 1.5. For a given value of g, the family given by Proposition 3.2.5 and Lemma 3.4.5 comprises a positive proportion of all hyperelliptic curves with a rational Weierstrass point, since the latter is defined by finitely many congruence conditions. By Corollary 3.2.3, at least 25% of the curves in this family have rank r ≤
Proof of Theorem 3. 2 With Lemma 3.7 at our disposal, we obtain a lower bound for the size of the set {1 ≤ n < x : G(n) ∈ P2}. We wish to apply Lemma 3.4 and Lemma 3.7 to equation (3.5) to obtain a lower bound for W (A, z). We may do this for each term in (3.5) but the short sum x1−s≤p<x λ/2 log x p X .0 − log p Σ S(A , z). However, in this case, we make the estimate Σ x S(Ap, z) p log(x/p), log x yielding the bound O . sx . For notational convenience, set log z α := 1 + γ0 and γ := . log x By partial summation, we obtain . .α Σ A γ Σ∫ 1 ∫ u u − t γ .α − u − t Σ dt du W ( , z) > V (z)x f + f γ γ 1 t t t u ∫ 1 . γ .α − u Σ .α − u ΣΣ du − γ (1 − 2u) u F + uF ∫ u γ u 1 − (1 − u) F .α − u Σ du Σ − sΣ 1 γ u =: V (z)x(W − s), where we have let λ tend to 2, which is permitted by continuity. Since ΓG ƒ= 0, we have that V (z) = log−1 x by Xxxxxxx’ Theorem and we wish to show that W > 0. We observe that W decreases monotonically as α increases from 1, so we wish to find γ < 1 such that W|α=1 > 0. However, we will not immediately substitute α = 1 into the above formula. Instead, we will choose γ = α and ∫ Σ take the limit as α tends to 1 from the right. Using that if 3 ≤ s ≤ 5, and sF (s) = 2eC .1 + s−1 du log(u − 1) u sf (s) = 2eC .log(s − 1) + ∫ s−1 ∫ t−1 log(u − 1) du dt Σ . . if 4 ≤ s ≤ 6, we obtain 3 2 u t αeC W = log 5 Σ
Proof of Theorem 3. 2 With Lemma 3.7 at our disposal, we obtain a lower bound for the size of the set {1 ≤ n < x : G(n) ∈ P2}. −
Proof of Theorem 3
