On the 1-3-5 conjecture and related topics

The 1-3-5 conjecture of Z.-W. Sun states that any $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as $x^2+y^2+z^2+w^2$ with $w,x,y,z\in\mathbb N$ such that $x+3y+5z$ is a square. In this paper, via the theory of ternary quadratic forms and related modular forms, we study the integer version of the 1-3-5 conjecture and related weighted sums of four squares with certain linear restrictions. Here are two typical results in this paper: (i) There is a finite set $A$ of positive integers such that any sufficiently large integer not in the set $\{16^ka:\ a\in A,\ k\in\mathbb N\}$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb Z$ and $x+3y+5z\in\{4^k:\ k\in\mathbb N\}$. (ii) Any positive integer can be written as $x^2+y^2+z^2+2w^2$ with $x,y,z,w\in\mathbb Z$ and $x+y+2z+2w=1$. Also, any sufficiently large integer can be written as $x^2+y^2+z^2+2w^2$ with $x,y,z,w\in\mathbb Z$ and $x+2y+3z=1$.