On restricted sums of four squares and Zhi-Wei Sun's $x+24y$ conjecture

In this paper, by using the arithmetic theory of ternary quadratic forms, we study some refinements on Lagrange’s four-square theorem. For example, given positive integers $a,b$ satisfying some algebraic conditions and a positive integer $C\ge 3$, we will show that for any sufficiently large integer $n$ with ${\text{ord}}_2(n)\le C$, there exist non-negative integers $x,y,z,w$ such that $$\{\begin{array}{l}{x}^2+y^2+z^2+w^2=n,ax+by\in \mathcal{S},\end{array}$$ where $\mathcal{S}$ is the set of all squares over $\mathbb{Z}$. In particular, we obtain some progress on Zhi-Wei Sun’s $x+24y$ conjecture.