Let k ≧ 2 be a fixed integer, A = {a1, a2, …} (a1 < a2 < …) be an infinite sequence of positive integers, and let Rk(n) denote the number of solutions of \documentclass{aastex}
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$$a_{i_1 } + a_{i_2 } + \cdots + a_{i_k } = n,a_{i_1 } \in \mathcal{A},...,a_{i_k } \in \mathcal{A}$$
\end{document}. Let B(A, N) denote the number of blocks formed by consecutive integers in A up to N. In [5], it was proved that if k > 2 and \documentclass{aastex}
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$$\lim _{N \to \infty } \frac{{B(\mathcal{A},N)}}{{\sqrt[k]{N}}}$$
\end{document} = ∞ then |δl(Rk(n))| cannot be bounded for l ≦ k. The aim of this paper is to show that the above result is nearly best possible. We are using probabilistic methods.