| MATHEMATICS OF COMPUTATION S 0025-5718(09)02211-X (2009) | |||||||||||||||
Abstract | |||||||||||||||
| Abstract. Let p/q be a rational noninteger number with p>q ≥ 2. A real number λ>0isaZ 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. 1. An approximate multiplication problem Let us begin with the following problem. Starting with a positive integer x, we consider the map ⎪ ⎨ 4x/3, if x ≡ 0 (mod 3), x → (4x +1)/3, if x ≡ 2 (mod 3), STOP, if x ≡ 1 (mod 3). Consider the iterates of this map, starting, for instance, with x =6. Wehave | |||||||||||||||
Details der Publikation | |||||||||||||||
| |||||||||||||||