Let k be a positive integer. Denote by \(D_{1/k}\) the smallest integer d such that for every set A of nonnegative integers with a lower density of 1/k, the set \((k + 1)A\) contains an infinite arithmetic progression with a difference of at most d, where \((k + 1)A\) refers to the set consisting of all the sums of \(k + 1\) elements (not necessarily distinct) of A. Chen and Li conjectured that \(D_{1/k}=k^2+o(k^2)\) . Subsequently, this conjecture was proved by Chen–Yang–Zhao, who established \(\begin{aligned} D_{1/k}=k^2+O\big (k^2 \exp (-c\sqrt{\log k})\big ), \end{aligned}\) where c is a positive constant. The purpose of this paper is to improve the error term by proving \(\begin{aligned} D_{1/k}=k^2+O\big (k^{\frac{19}{10}+\frac{1}{10}\times 10^{-10}}\log ^7 k\big ). \end{aligned}\)
