Hey, having majority of culture in a sector gives a lot of prestige points. This can be used, not only to get ahead, but also to throw someone down from the first place.
You can push their culture either through conquest, or culture starbases.
Conquest overall is a great tool to hinder score of your ai opponents, as it takes away points in fleet strenght, production, etc.
Just make sure to win, and not lose in a war 
