[math-fun] How can I accurately calculate (p/q)^n mod 1, for large n

Short of using a multi-precision package, is there some hack for calculating high powers of a rational number mod 1, to at least 3 decimal places (but ideally more like 10). Mma seems to be able to do one such calculation quickly, but takes forever, if not longer, in a large loop. So I'm wondering about shortcuts. --Dan Sometimes the brain has a mind of its own.