The most basic distributed radio network broadcasting primitive - called Decay - dates back to a PODC'87 result of Bar-Yehuda, Goldreich, and Itai. In any radio network with some informed source nodes, running Decay for O(d \log n + \log^2 n) rounds informs all nodes at most $d$ hops away from a source with high probability. Since 1987 this primitive has been the most important building block for implementing many other functionalities in radio networks. The only improvements to this decades-old algorithm are slight variations due to [Czumaj, Rytter; FOCS'03] and [Kowalski and Pelc, PODC'03] which achieve the same functionality in O(d \log \frac{n}{d} + \log^2 n) rounds. To obtain a speedup from this, $d$ and thus also the network diameter need to be near linear, i.e., larger than n^{1-\eps}.
Our new distributed primitive spreads messages for $d$ hops in O(d \frac{\log n \log \log n}{\log d} + \log^{O(1)} n) rounds. This improves over Decay for any super-polylogarithmic d = \log^{\omega(1)} n and achieves a near-optimal O(d \log \log n) running time for d = n^{\eps}. This also makes progress on an open question of Peleg.