Let p/q be a rational noninteger number with p > q ≥ 2. A real number λ > 0 is a Zp/q-number if {λ(p/q)n} < 1/q for every nonnegative integer n, where {x} denotes the fractional part of x. We develop several algorithms to search for Zp/q-numbers, and use them to determine lower bounds on such numbers for several p and q. It is shown, for instance, that if there is a Z3/2-number, then it is grea...