In this paper, we study a density version of Waring’s problem involving \( s_{1} \) squares and \( s_{2} \) k-th powers, where \( k\ge 3 \) is an odd integer. Assume that A is a positive density subset of squares and B is a positive density subset of k-th powers. If the lower densities of A and B satisfy certain conditions, then for all sufficiently large positive integers n, there exist \( x_{1}^{2},\ldots ,x_{s_{1}}^{2}\in A \) and \(y_{1}^{k},\ldots ,y_{s_{2}}^{k} \in B\) such that \(n=x_{1}^{2}+\cdots +x_{s_{1}}^{2}+y_{1}^{k}+\cdots +y_{s_{2}}^{k}\) . In particular, when \(s_{1}=1\) , we obtain a density version of Waring’s problem for one square and \(s_{2}\) k-th powers. The study of this type of Diophantine equation has a long history. The same method can be generalized to deduce analogous results for other equations involving mixed powers of higher degrees. Additionally, due to some technical difficulties in handling mixed powers, we are limited to restricting k to be odd.
