Restricted sums of four squares

Abstract: We refine Lagrange's four-square theorem in new ways by imposing some restrictions involving powers of two (including $1$). For example, we show that each $n=1,2,3,\ldots$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb N={0,1,2,\ldots})$ with $|x+y-z|\in{4^k:\ k\in\mathbb N}$ (or $|2x-y|\in{4^k:\ k\in\mathbb N}$, or $x+y-z\in{\pm 8^k:\ k\in\mathbb N}\cup{0}\subseteq{t^3:\ t\in\mathbb Z}$), and that we can write any positive integer as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb Z)$ with $x+y+2z$ (or $x+2y+2z$) a power of four. We also prove that any $n\in\mathbb N$ can be written as $x^2+y^2+z^2+2w^2$ $(x,y,z,w\in\mathbb Z)$ with $x+y+z+w$ a square (or a cube). In addition, we pose some open conjectures for further research; for example, we conjecture that any integer $n>1$ can be written as $a^2+b^2+3^c+5^d$ with $a,b,c,d\in\mathbb N$.