The phrase "gat someone case" means to scrutinize or investigate someone closely. There are several synonyms for this phrase that can be used in different contexts. One of the synonyms is "monitor," which means to keep a check on someone's activities. Another synonym is "surveillance," which entails observing someone secretly. "Shadowing" is another synonym that refers to following someone closely to gather information about them. "Stalking" is another synonym, but it has negative connotations, as it refers to obsessively following someone and making them feel unsafe. Lastly, "watching closely" is a simpler synonym that means to keep a close eye on someone.