Let N 0 be the set of all nonnegative integers and ? ? 2 be a fixed integer. For A ? N 0 and n ? N 0 , let r ? ' ( A , n ) denote the number of solutions of a 1 + ? + a ? = n with a 1 , ? , a ? ? A and a 1 ? ? ? a ? . Let k be a fixed positive integer. In this paper, we prove that, for any given distinct positive integers u i ( 1 ? i ? k ) and positive rational numbers α i ( 1 ? i ? k ) with α 1 + ? + α k = 1 , there are infinitely many sets A ? N 0 such that r ? ' ( A , n ) ? 1 for all n ? 0 and the set of n with r ? ' ( A , n ) = u i has density α i for all 1 ? i ? k .
['Yong-Gao Chen', 'Min Tang']
On a generalization of a theorem of Sárközy and Sós