Take a negative rational number -p/q and raise it to successive powers. The integer parts oscillate in sign. The fractional parts — the remainders after subtracting the nearest integer — form a sequence that clusters in specific regions of the interval [0,1].

How small can those clustering intervals be? This paper proves sharp bounds: the fractional parts of (-p/q)^n cannot all lie in an interval shorter than a critical threshold determined by p and q. Below that threshold, some power must escape the interval. The bound is tight — there exist starting values that achieve it.

The proof connects to dynamics on the real line. Powers of a rational number generate an orbit under multiplication, and the fractional parts are this orbit viewed modulo 1. The clustering question becomes: how concentrated can a multiplicative orbit be on the unit interval? The answer depends on the arithmetic properties of the base — specifically, the greatest common divisor structure of p and q.

What makes the negative case different from the positive is the sign alternation. Positive powers of p/q always grow in the same direction. Negative powers alternate, and the fractional parts inherit this oscillation. The oscillation doesn’t help with clustering — it constrains it differently. The orbit visits both sides of 1/2 regularly, which limits how tightly it can concentrate in any subinterval.

A question about arithmetic becomes a question about dynamics. The number’s rationality constrains its orbit’s geometry. Irrationals can cluster arbitrarily tightly; rationals cannot.