For an integer
![$ b\geq 2$](abs/img2.gif)
and for
![$ x\in [0,1)$](abs/img3.gif)
, define
![$ \rho_b(x) = \sum_{n=0}^{\infty} \frac{\{\mskip -5mu\{b^nx\}\mskip -5mu\}}{b^n}$](abs/img4.gif)
,
where
![$ \{\mskip -5mu\{t\}\mskip -5mu\}$](abs/img5.gif)
denotes the fractional part of the real number
![$ t$](abs/img6.gif)
.
A number of
properties of
![$ \rho_b$](abs/img7.gif)
are derived, and then a connection between
![$ \rho_b$](abs/img7.gif)
and the rumor
conjecture is established. To form a rumor sequence
![$ \{z_n\}$](abs/img8.gif)
, first select integers
![$ b\geq 2$](abs/img2.gif)
and
![$ k\geq 1$](abs/img9.gif)
. Then
select an integer
![$ z_0$](abs/img10.gif)
, and for
![$ n\geq 1$](abs/img11.gif)
let
![$ z_n = bz_{n-1} \bmod{(n+k)}$](abs/img12.gif)
, where the right
side is the least non-negative residue of
![$ bz_{n-1}$](abs/img13.gif)
modulo
![$ n+k$](abs/img14.gif)
. The rumor sequence
conjecture asserts that all such rumor sequences are eventually 0. A condition on
![$ \rho_b$](abs/img7.gif)
is shown to be equivalent to the rumor conjecture.
Received December 1 2010;
revised version received February 1 2011.
Published in Journal of Integer Sequences, February 19 2011.