Recently, interest has increased in applying reactive synthesis to richer-than-Boolean domains. A major (undecidable) challenge in this area is to establish when certain repeating behaviour terminates in a desired state when the number of steps is unbounded. Existing approaches struggle with this problem, or can handle at most deterministic games with Büchi goals. This work goes beyond by contributing the first effectual approach to synthesis with full LTL objectives, based on Boolean abstractions that encode both safety and liveness properties of the underlying infinite arena. We take a CEGAR approach: attempting synthesis on the Boolean abstraction, checking spuriousness of abstract counterstrategies through invariant checking, and refining the abstraction based on counterexamples. We reduce the complexity, when restricted to predicates, of abstracting and synthesising by an exponential through an efficient binary encoding. This also allows us to eagerly identify useful fairness properties. Our discrete synthesis tool outperforms the state-of-the-art on linear integer arithmetic (LIA) benchmarks from literature, solving almost double as many syntesis problems as the current state-of-the-art. It also solves slightly more problems than the second-best realisability checker, in one-third of the time. We also introduce benchmarks with richer objectives that other approaches cannot handle, and evaluate our tool on them.