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30 Chapter 2. On a conjecture about addition modulo 2 k − 12.3.2 MeanFor t ≠ t ′ , S t,k ∩ S t′ ,k = ∅, so thatS ==2⊔k −2t=0S t,k{(a, b) ∈ ( Z/(2 k − 1)Z ) }2| wH (a) + w H (b) ≤ k − 1,and summing up according to the value of w H (a) + w H (b), we computek−1∑( ) 2k#S =ii=0= 2 2k−1 − 1 ( ) 2k.2 kThe following proposition shows that the bound of the conjecture is sharp.Proposition 2.3.3. For k ≥ 2,(E t (#S t,k ) = 2 k−1 1 − √ 1 ( )) 1+ o √kπk.Proof. Using Stirling’s approximation, we have( 2kk)= 2k!k! 2 ∼ 22k√πk,and we computeE t (#S t,k ) = #S2 k − 1= 22k−1 − 2( 1 2kk2 k − 12 2k2 2k−1 − 1 √2 πk+ o=2 k − 1= 22k−12 k − 1)( )2√ 2kk(1 − 1 √πk+ o( 1 √k))= 2 k−1 (1 + 1 2 k + o ( 12 k )) (1 − 1 √πk+ o(= 2 k−1 1 − √ 1 ( )) 1+ o √kπk.( 1 √k))2.3.3 ZeroWe now deal with the pathological case t = 0.Proposition 2.3.4. For k ≥ 2,S 0,k = {(0, 0)} .