The optimal scale in Linnik's microsquare problem for almost all integers

We prove that, for any positive function $\mathrm{\Psi }(t)\to \mathrm{\infty }$, almost all positive integers $n$ satisfying $4\nmid n$ and $n\equiv ̸7(\mathrm{m}\mathrm{o}\mathrm{d}8)$ can be written as $n=x^2+y^2+z^2$, where $x,y,z\in \mathbb{Z}$ and $1⩽z<\sqrt{\log n}\mathrm{\Psi }(n)$. This improves Wooley’s bound, and we show that the result is best possible. Our approach combines Hooley’s weighted representation function for sums of two squares with Green’s $W$-trick. We also give a necessary and sufficient condition, in terms of covering systems, for almost all odd integers to be represented as the sum of a squarefree number and a power of two, answering a question of Granville and Soundararajan.