On the Additive Complexity of Some Integer Sequences

Abstract

The paper presents several results concerning the complexity of calculations in the model of vector addition chains. A refinement of N. Pippenger’s upper bound is obtained for the complexity of the class of integer \(m \times n\) matrices with the constraint \(q\) on the size of the coefficients as \(H=mn\log_2 q \to \infty\) up to \(\min\{m,n\}\log_2 q+(1+o(1))H/\log_2 H+n\). Next, we establish an asymptotically tight bound \((2+o(1))\sqrt n\) on the complexity of сomputation of the number \(2^n-1\) in the base of powers of \(2\). Based on generalized Sidon sequences, constructive examples of integer sets of cardinality \(n\) are constructed: sets, with polynomial size of elements, having the complexity \(n+\Omega(n^{1-\varepsilon})\) for any \(\varepsilon>0\) and sets, with the size \(n^{O(\log n)}\) of the elements, having the complexity \(n+\Omega(n)\).