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2.3. The case ɛ = +1 33Theorem 2.3.14. Let k ≥ 2 and t ∈ ( Z/(2 k − 1)Z ) ∗. If t ≃ −t, then#S t,k ≤ 2 k−1 − 1if 2t ≠ −t, and#S t,k ≤ 2 k−1otherwise.Proof. t ≃ −t so that #S −t,k = #S t,k . If 2t ≠ −t, then Corollary 2.3.13 becomes#S t,k ≤ 2 k−1 − 1 2 .But #S t,k ∈ N, so that the following inequality holds:#S t,k ≤ 2 k−1 − 1 .If 2t = −t, then only the following one is true:#S t,k ≤ 2 k−1 .2.3.6 A combinatorial proposition of independent interestThe following quantities may be used to study #S t,k .Definition 2.3.15. Let d and n be positive integers and• Σ(d, n) = ∑ nl=0 2−l( )l+dd ,• ∆(d, n) = 2 −n( )n+d+1 d−nd 2d+2 .They are related through the following combinatorial identity.Proposition 2.3.16. For any d, n and e positive integers,e∑Σ(d + e, n + e) = 2 e Σ(d, n) + 2 e−l ∆(d + l − 1, n + l − 1) .l=1