Before revision:  Specifically, for prescribed parameters \vartheta,\varepsilon \in (0,1), the computed spanner G has O (n \bigl( \varepsilon^{-c} \vartheta^{-6} \log^{c\bigr) edges, where c= O(d).
After revision:  Specifically, for prescribed parameters \vartheta,\varepsilon \in (0,1), the computed spanner G has O (n \bigl( \varepsilon^{-c} \vartheta^{-6} n \log n (\log\log n)^6\bigr) edges, where c= O(d).