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70 Chapter 2. On a conjecture about addition modulo 2 k − 1Finally, if d = 1, then ( )d−1+jd−1 = 1, so that the sum becomesand, if d = 2, then ( )d−1+jd−1 = j + 1, so thatP 2 = 1 4 2P 1 = 1 14 1 − 1/4 = 1 3 ;∞ ∑j=0(j + 1) 24 j = 1 4(= 1 214( 2)4 1 −1 3+4= 2 27 + 1 9 = 527 .14∞∑j=0(1 −14When the number of blocks d goes as well to infinity, f d (∞, . . . , ∞) converges towards 1/2.Indeed 1 ≤ P3 d d ≤ 1 + 3 14 d 4converges towards 0 as d goes to infinity. As we show below, it does2 dso monotonically so that f d (∞, . . . , ∞) goes to 1/2 monotonically as well.A first step towards proving the monotonicity of P d in d is to study the special case d = 1. Inthis case the value P [X 1 = k] has indeed a short closed-form expression.j 24 j) 2)Lemma 2.6.6. For d = 1,Proof. Indeed, for k ≥ 0,P [X 1 = k] = 13 · 2 |k| .P [X 1 = k] = P [G 1 = k + H 1 ]∞∑= P [G 1 = i] P [H 1 = k + i]=i=0∞∑i=01 12 i+1 2 k+i+1 = 1 ∑∞ 12 k+2 4 i= 1 42 k+2 3 = 1 312 k .In the general case d ≥ 1, it can also be proven quite directly that the maximal value ofP [X d = k] is attained for k = 0.Lemma 2.6.7. For d ≥ 1 and k ≠ 0,P [X d = k] < P [X d = 0] .Proof. Consider the real Hilbert space H = l 2 (Z, R) of square-summable sequences. It is equippedwith norm preserving translation operators τ k defined by (τ k u) j = u j+k for a sequence u =(u j ) j∈Z ∈ H. Consider now the sequence u (d) ∈ H defined byu (d)ji=0[ d∑] [ d∑]= P G i = j = P H i = ji=1i=1whose exact values are given in Lemma 2.6.4 for j ≥ 0, and u (d)j = 0 for j < 0.