An oscillation theorem on the additive representative function over \(\mathbb{N}\) (Q784711)

Let \(A\) be an infinite non-empty subset of \(\mathbb{N}\). For each \(n\in \mathbb{N}\), define \[r_{A,A}(n):=|\{(a, b): a, b\in A, a + b = n\}|\] and \[R_{A,A}(n):= \sum \limits_{j\leq n} r_{A,A}(j).\] The two authors show that if the function \(R_{A,A}(n)\) is well-distributed in some sense, then it cannot be very well distributed. Explicitly, if for some constant \(c>0\), \[\limsup\limits_{n\to \infty}\frac{|R_{A,A}(n)-cn|}{n^{\frac{1}{4}}}<+\infty\] then for some constant \(\delta> 0\), the set \[\{n\in \mathbb{N}: |R_{A,A}(n)-cn|\geq \delta n^{\frac{1}{4}}\}\] has a positive lower density. The nice result implies the well-known Erdős-Fuchs theorem. A similar generalization for multi-sum is obtained, too.