Inverse Erdős-Fuchs theorem for \(k\)-fold sumsets (Q401971)

The authors generalize a result of \textit{I. Z. Ruzsa} [J. Number Theory 62, No. 2, 397--402 (1997; Zbl 0872.11014)] on the inverse Erdős-Fuchs theorem for \(k\)-fold sumsets. The following theorem is proved:NEWLINENEWLINE NEWLINETheorem 1.1. Suppose that \(k\geq 2\) is an integer and \(\beta< k\) is a positive real number. Then there exists a sequence \(A = \{a_1\leq a_2\leq a_2\leq \ldots\}\) of positive integers, satisfyingNEWLINENEWLINE\[NEWLINE\sum_{m\leq n} r_{kA}(m)-Cn^\beta=\begin{cases} O\left(n^{\beta-\beta(k+\beta)/k^2}\sqrt{\log n}\right),\quad &\text{if}\;k > 2\beta,\\ NEWLINEO\left(n^{\beta-3\beta/(2k)}\sqrt{\log n}\right),\quad &\text{if}\;k < 2\beta,\\NEWLINEO\left(n^{\beta-3/4} \log n\right),\quad &\text{if}\;k = 2\beta,\end{cases}NEWLINE\]NEWLINE NEWLINEwhere \(C\) is a constant.