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For any constants $d\ge 1$, $\epsilon >0$, $t>1$, and any $n$-point set$P\subset\mathbb{R}^d$, we show that there is a geometric graph $G=(P,E)$having $O(n\log^2 n\log\log n)$ edges with the following property: For any$F\subseteq P$, there exists $F^+\supseteq F$, $|F^+| \le (1+\epsilon)|F|$ suchthat, for any pair $p,q\in P\setminus F^+$, the graph $G-F$ contains a pathfrom $p$ to $q$ whose (Euclidean) length is at most $t$ times the Euclideandistance between $p$ and $q$.  In the terminology of robust spanners (Bose \et al, SICOMP, 42(4):1720--1736,2013) the graph $G$ is a $(1+\epsilon)k$-robust $t$-spanner of $P$. Thisconstruction is sparser than the recent constructions of Buchin, Ol\`ah, andHar-Peled (arXiv:1811.06898) who prove the existence of $(1+\epsilon)k$-robust$t$-spanners with $n\log^{O(d)} n$ edges.

Near-Optimal $O(k)$-Robust Geometric Spanners