Solubility of dimethyl ether in aliphatic hydrocarbons
Since dimethyl ether is a very uncommon solvent, it's not a big surprise that you have hard time finding its information just using Google. However, I found two relatively important parameters of dimethyl ether: Its Hildebrand Solubility Parameter is 7.3 and Kauri-Butanol index is 90 (INVENTEC). For comparision, Hildebrand Solubility Parameter of diethyl ether is 7.62 (Solubility Parameters: Theory and Application). According to this reference, if two solvents have about the same Hildebrand values, theoretically, such solvents should have solubility behaviors similar to each other. 

Thus, dimethyl ether should behave similar to diethyl ether (in liquid phase, I believe).  Keep in mind that most of hydrocarbons dissolve in diethyl ether in all ratios. For example, we run column chromatography using such solvent systems (diethyl ether/hexanes) to isolate close nonpolar organic compounds from complex mixtures.