Lower bounds for Z-numbers

Let p/q be a rational noninteger number with p > q ≥ 2. A real number λ ≥ 0 is a Z p/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 Z p/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 Z 3/2 -number, then it is greater than 2 57 . We also explore some connections between these problems and some questions regarding iterated maps on integers.