LOWER BOUNDS FOR Z-NUMBERS

Let p/q be a rational noninteger number with p > q >= 2. A real number lambda > 0 is a Z(p/q)-number if {lambda(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.