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For an integer b ≥ 2 and for x ∈ [0, 1), define ρ b (x) = ∞ n=0 {{b n x}} b n , where {{t}} denotes the fractional part of the real number t. A number of properties of ρ b are derived, and then a connection between ρ b and the rumor conjecture is established. To form a rumor sequence {z n }, first select integers b ≥ 2 and k ≥ 1. Then select an integer z 0(More)
Rumor sequences are generated recursively as follows: fix nonnegative integers b, c, and let z 0 = 0. For n ≥ 1, define z n = (bz n−1 + c) mod n, where the least nonnegative residue modulo n is taken. There have been a few papers dealing with the behavior of rumor sequences, but they all concern that behavior when the value of c is fixed. It turns out that(More)
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