On the distance from a rational power to the nearest integer

✍ Scribed by Artūras Dubickas

• Publisher: Elsevier Science
• Year: 2006
• Tongue: English
• Weight: 222 KB
• Volume: 117
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2005.07.004

✦ Synopsis

We prove that for any non-zero real number the sequence of fractional parts { (3/2) n }, n = 1, 2, 3, . . . , contains at least one limit point in the interval [0.238117 . . . , 0.761882 . . .] of length 0.523764 . . . . More generally, it is shown that every sequence of distances to the nearest integer || (p/q) n ||, n = 1, 2, 3, . . . , where p/q > 1 is a rational number, has both 'large' and 'small' limit points. All obtained constants are explicitly expressed in terms of p and q. They are also expressible in terms of the Thue-Morse sequence and, for irrational , are best possible for every pair p > 1, q = 1. Furthermore, we strengthen a classical result of Pisot and Vijayaraghavan by giving similar effective results for any sequence || n ||, n = 1, 2, 3, . . . , where > 1 is an algebraic number and where = 0 is an arbitrary real number satisfying / ∈ Q( ) in case is a Pisot or a Salem number.