Before revision:  In this work we improve the latter result by devising a simple deterministic algorithm for a +(6+\varepsilon)W spanner for weighted graphs with size O(n^{4/3}) (for any constant \varepsilon>0), thus nearly matching the classical +6 spanner of size O(n^{4/3}) for unweighted graphs.  ]  We also show a simple randomized algorithm for a +4W emulator of size O(n^{4/3})  \cdot W) spanner, and we also obtain a tradeoff between size and stretch.
After revision:  In this work we improve the latter result by devising a simple deterministic algorithm for a +(6+\varepsilon)W spanner for weighted graphs with size O(n^{4/3}) (for any constant \varepsilon>0), thus nearly matching the classical +6 spanner of size O(n^{4/3}) for unweighted graphs. ] We also show a simple randomized algorithm for a +4W emulator of size O(n^{4/3}) . In addition, we show that our technique is applicable for very sparse additive spanners, that have linear size. For weighted graphs, we use a variant of our simple deterministic algorithm that yields a linear size + \cdot W) spanner, and we also obtain a tradeoff between size and stretch.