On the 1-3-5 conjecture and related topics

The 1-3-5 conjecture of Z.-W. Sun states that any $n\in\mathbb N={0,1,2,\ldots}$canbewrittenas$x^2+y^2+z^2+w^2$with$w,x,y,z\in\mathbb N$suchthat$x+3y+5z$ 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 $\{{16}^{k}a:a\in A,k\in \mathbb{N}\}$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in \mathbb{Z}$ and $x+3y+5z\in{4^k:\ k\in\mathbb N}$\mathrm{.}(ii)Anypositiveintegercanbewrittenas$x^2+y^2+z^2+2w^2$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 $x+2y+3z=1$.