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32 Chapter 2. On a conjecture about addition modulo 2 k − 1We introduce the following notation to exclude the special pairs (a, b) involving zero.Definition 2.3.10. For k ≥ 2 and t ∈ ( Z/(2 k − 1)Z ) ∗, we define S∗t,kasS ∗ t,k = S t,k \ {(0, t), (t, 0)} .We can now relate T t,k and S −t,k either through negation, or through translation.Proposition 2.3.11. For k ≥ 2 and t ∈ ( Z/(2 k − 1)Z ) ∗,T t,k = −S ∗ −t,k .Proof. Indeed, if (a, t − a) ∈ T t,k , then a ≠ 0, a ≠ t and r(−a, t) < w H (t), so that r(a, −t) >w H (−t) and (−a, −t + a) ∈ S ∗ −t,k .Conversely if (a, −t − a) ∈ S ∗ −t,k , then (−a, t + a) ∈ T t,k.Proposition 2.3.12. For k ≥ 2 and t ∈ ( Z/(2 k − 1)Z ) ∗,T t,k = t + S ∗ −t,k .Proof. If (a, −t − a) ∈ S −t,k and a ≠ 0, −t, thenMoreoverr(−a, −t) = w H (−a) + w H (−t) − w H (−a − t) < w H (−t) = k − w H (t) .r(−t − a, t) = w H (−t − a) + w H (t) − w H (−t − a + t)= w H (−t − a) + (k − w H (−t)) − w H (−a)= k − r(−a, −t) ,so that r(−t − a, −t) > w H (t) and t + (a, −t − a) ∈ T t,k .Conversely, if (a, t − a) ∈ T t,k , then a ≠ 0, t and a − t ∈ S ∗ −t,k .We could also have used the swap function and the previous corollary.These relations then relate S t,k and S −t,k .Corollary 2.3.13. Let k ≥ 2. If 2t ≠ −t, then#S t,k + #S −t,k ≤ 2 k − 1 .Otherwise#S t,k + #S −t,k ≤ 2 k .Proof. We already know that S t,k ⊔ T t,k ⊂ C t,k so that #S t,k + #S −t,k ≤ 2 k + 1. In factw H (t + t) = w H (2t) = w H (t) so that (2t, −t) and (−t, 2t) are in E t , i.e. neither in S t,k nor in T t,k .Finally{2#S t,k + #S −t,k ≤k − 1 if 2t ≠ −t ,#S t,k + #S −t,k ≤ 2 k if 2t = t .We can now prove the conjecture in the very specific case where t ≃ −t.