An Update on the 3x+1 Problem Marc Chamberland Contents

12
and Lagarias[8](1995). The functions e l (k, ·) are integrable with respect to Z ∗ 3 ’s
unique normalized Haar measure, yielding the 3-adic average
⎛ ⎞
∫
1
ē l (k) := e l (k, a)da = ⎝ k + l ⎠
2 · 3 l−1 .
l
Several results follow with this approach, including
∫
1
lim inf
n→∞ 2 n s n (a)da > 0,
Z ∗ 3
Z ∗ 3
implying that every predecessor set P T (a) with a ≢
density.
0 mod 3 has positive
3 Representations of Iterates of a 3x + 1 Map
The oft-cited result in this area is due to Böhm and Sontacchi[14](1978): the
3x + 1 problem is equivalent to showing that each n ∈ Z + may be written as
(
)
n = 1
m−1
∑
3 m 2 vm − 3 m−k−1 2 v k
where m ∈ Z + and 0 ≤ v 0 < v 1 < · · · v m are integers. Similar results were
obtained by Amigó[2, Prop.4.2](2001).
k=0
Sinai[70](2003) studies the function F on the set ⊓ defined as
⊓ = {1} ∪ ⊓ + ∪ ⊓ − , ⊓ ±1 = 6Z + ± 1.
Let k 1 , k 2 , . . . , k m ∈ Z + and ɛ = ±1. Then the set of x ∈ ⊓ ɛ to which one can
apply F (k1) F (k2) · · · F (km) is an arithmetic progression
σ (k1,k2,...,km,ɛ) = {6 · (2 k1+k2+···+km + q m ) + ɛ}
for some q m such that 1 ≤ q m ≤ 2 k1+k2+···+km . Moreover,
F (k1) F (k2) · · · F (km) (σ (k1,k2,...,km,ɛ) ) = {6 · (3 m p + r) + δ}
for some r and δ satisfying 1 ≤ r ≤ 3 m , δ = ±1. Results in this spirit may
be found in <strong>An</strong>drei et al.[5](2000) and Kuttler[44](1994). Sinai uses this result