On a generalization of a theorem of Sárközy and Sós

Let N0N0 be the set of all nonnegative integers and ℓ≥2ℓ≥2 be a fixed integer. For A⊆N0A⊆N0 and n∈N0n∈N0, let rℓ′(A,n) denote the number of solutions of a1+⋯+aℓ=na1+⋯+aℓ=n with a1,…,aℓ∈Aa1,…,aℓ∈A and a1≤⋯≤aℓa1≤⋯≤aℓ. Let kk be a fixed positive integer. In this paper, we prove that, for any given distinct positive integers ui(1≤i≤k) and positive rational numbers αi(1≤i≤k) with α1+⋯+αk=1α1+⋯+αk=1, there are infinitely many sets A⊆N0A⊆N0 such that rℓ′(A,n)≥1 for all n≥0n≥0 and the set of nn with rℓ′(A,n)=ui has density αiαi for all 1≤i≤k1≤i≤k.