The phrase "serious inroads" means making significant progress or gaining strong influence in a particular area, often by overcoming obstacles or competing successfully with others. Full definition
The other two ideas are obviously capable of making serious inroads into employee protection. (newlawjournal.co.uk)
Citing the iPod Touch as making serious inroads into the handheld market, he said shared his thoughts on the PSP2's chances for success: (playstationlifestyle.net)
However, all these foreign companies have failed to any make serious inroads in discovering bauxite in commercial quantities in order to build an Integrated Aluminum Industry. (ghanapoliticsonline.com)