"To dig in" is an idiomatic expression that denotes the act of becoming firmly established or showing determination. In terms of synonyms, there are a variety of phrases that can convey a similar meaning. For instance, "to hunker down" signifies committing oneself fully to a task or situation. Similarly, "to buckle down" implies an intensified focus and dedication. Another alternative is "to knuckle down", which emphasizes the necessity of hard work and concentration. Additionally, one could use "to get serious", which denotes a shift towards a more resolute and concentrated approach. All these phrases can be employed to express the notion of firmly applying oneself to a task or becoming deeply involved in a situation, rather than merely superficially engaging with it.