Blowups with log canonical singularities

We show that the minimum weight of a weighted blow-up $\mathbf A^d$ with $\varepsilon$-log canonical singularities is bounded by constant depending only on $\varepsilon $ and $d$. This was conjectured Birkar. Using recent classification $4$-dimensional empty simplices Iglesias-Vali\~no Santos, we work out an explicit bound for blowups A^4$ terminal singularities: smallest always at most $32$, $6$ in all but finitely many cases.