A Function Related to the Rumor Sequence Conjecture

For an integer $b\geq 2$ and for $x\in [0,1)$ , define $\rho_b(x) = \sum_{n=0}^{\infty} \frac{\{\mskip -5mu\{b^nx\}\mskip -5mu\}}{b^n}$ , where $\{\mskip -5mu\{t\}\mskip -5mu\}$ denotes the fractional part of the real number

. A number of properties of $\rho_b$ are derived, and then a connection between $\rho_b$ and the rumor conjecture is established. To form a rumor sequence $\{z_n\}$ , first select integers $b\geq 2$ and $k\geq 1$ . Then select an integer

, and for $n\geq 1$ let $z_n = bz_{n-1} \bmod{(n+k)}$ , where the right side is the least non-negative residue of $bz_{n-1}$ modulo

. The rumor sequence conjecture asserts that all such rumor sequences are eventually 0. A condition on $\rho_b$ is shown to be equivalent to the rumor conjecture.

A Function Related to the Rumor Sequence Conjecture

Bruce Dearden, Joel Iiams, and Jerry Metzger University of North Dakota Department of Mathematics Witmer Hall Room 313 101 Cornell Street Stop 8376 Grand Forks, ND 58202-8376 USA

Bruce Dearden, Joel Iiams, and Jerry Metzger
University of North Dakota
Department of Mathematics
Witmer Hall Room 313
101 Cornell Street Stop 8376
Grand Forks, ND 58202-8376
USA