REVIEW 4 major objections 5 minor 23 references
Higher dimensional shrinking target problem in beta dynamical systems
T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read This paper proves an exact Hausdorff dimension formula for two-dimensional shrinking target sets in beta-dynamical systems: the dimension is the smaller of two pressure-function roots.
desk verdict Genuinely new two-dimensional shrinking-target formula for beta-systems, but the lower-bound proof leans on an unproved transferred estimate and a missing separation reduction; worth refereeing, not ready as is. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The engine of the proof is the topological pressure $P(\phi)=\lim_{n\to\infty}\frac1n\log\sum_{(\epsilon_1,\dots,\epsilon_n)\in\Sigma_\beta^n}\sup_{y\in I_n}e^{S_n\phi(y)}$, whose roots encode the critical exponent of the covering series. Around it, the paper uses the theory of full cylinders, meaning cylinders of maximal length $\beta^{-n}$, together with two structural facts: an interval of length $\beta^{-l}$ can be covered by $O(l)$ cylinders of order $l$, and inside any sufficiently small interval one can find a full cylinder of length comparable to a prescribed power of the interval's length. These facts let the authors place full cylinders inside the shrinking target balls and assemble them into a Cantor set $F_\infty$. A measure is then assigned level by level, with exponents $s_i$ converging to the pressure root $s_0$; the mass distribution principle converts the measure estimate $\mu(I)\le |I|^{s/(1+\epsilon)}$ into the lower bound on the Hausdorff dimension.
What would settle it
Set $\beta=2$ and $f=g=1$; the theorem predicts $\dim_H E(T_2,1,1)=2\log 2/(\log2+1)\approx0.819$. Because the doubling map is a full shift, the Cantor construction is explicit, so a direct cylinder-counting computation of the Hausdorff dimension of the limsup set of rectangles of side $2^{-n(1+1/\log2)}$---or a numerical box-counting estimate---would confirm or refute the formula.
Extended reading notes
Core claim
The central result is Theorem 1.1: for $f,g$ positive continuous on $[0,1]$ with $f(x)\ge g(y)$ for all $x,y$, the shrinking-target set $E(T_\beta,f,g)$ has Hausdorff dimension \[ \dim_H E(T_\$\beta$,f,g)=\min\{s_1,s_2\}, \] where $s_1=\inf\{s\ge0:P(f-s(\log\beta+f))+P(-g)\le0\}$ and $s_2=\inf\{s\ge0:P(-s(\log\beta+g))+\log\beta\le0\}$. Here $P$ is the topological pressure of the $\beta$-dynamical system. The dimension is thus the smaller root of two pressure equations, each attached to a different covering strategy for the rectangles that appear in the natural limsup representation of the set. The upper bound follows by counting covers; the lower bound is obtained by constructing a Cantor subset built from full cylinders, defining a measure whose exponents converge to the pressure root, and applying the mass distribution principle.
Load-bearing premise
In Section 4.4.1, Case I, Step 3, the lower-bound proof invokes an estimate, stated as following 'with similar arguments' from [18] and [22], that bounds the total measure of all possible continuations of a cylinder by $\beta^{l\epsilon}$; if that estimate does not carry over to the $\beta_N$-approximating subshift used in the Cantor construction, the lower bound for the Hausdorff dimension does not follow.
Editorial extensions
If this is right
- The Hausdorff dimension of $E(T_\beta,f,g)$ depends only on $\beta$, $f$, and $g$, not on the target point $(x_0,y_0)$.
- The critical exponent is sharp: for $s>\min\{s_1,s_2\}$ the natural covering series converges and the $s$-dimensional Hausdorff measure is zero, while for $s<\min\{s_1,s_2\}$ the Cantor construction supplies a positive lower bound.
- Which covering strategy wins is decided by the size of $\min\{s_1,s_2\}$: if it exceeds $1$, covering by the shorter side of the rectangle is efficient; if it is at most $1$, covering by the longer side governs the dimension.
- For constant potentials the formula becomes explicit; with $\beta=2$ and $f=g=1$, it gives $\dim_H E(T_2,1,1)=2\log 2/(\log2+1)\approx0.819$.
Reading between the lines
- The lower-bound construction is written under the stronger hypothesis $f\ge(1+\epsilon)g$, while the theorem states only $f\ge g$; the paper leaves implicit the approximation step that would reduce one to the other, so a continuity check of the pressure formula near $g=f$ would show whether the stronger hypothesis is necessary.
- The same two-pressure-root mechanism should extend to weighted shrinking targets: replacing the single ordering $f\ge g$ by coordinate weights should replace the minimum over two roots by a minimum over more pressure roots.
- For $\beta$ a simple Parry number such as the golden ratio, admissibility has a finite lexicographic description, so the unproved continuation estimate from Step 3 can be checked by finite computation; this would test the transfer from $\beta_N$-subshifts outside the full-shift case $\beta=2$.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the two-dimensional shrinking target problem for beta-transformations with point-dependent approximation functions f,g satisfying f >= g. It claims that the Hausdorff dimension of the limsup set E(T_beta, f, g) equals min{s1, s2}, where s1 and s2 are defined as infima of pressure-function inequalities. The proof splits into an upper bound, obtained by natural covers and pressure series, and a lower bound, attempted via a Cantor set construction and the mass distribution principle. The upper bound is straightforward and essentially standard. The lower bound is the main substance but, as written, relies on an unjustified separation assumption, an unproved and index-inconsistent estimate delegated to earlier papers, and an unsubstantiated containment of the constructed Cantor set in the target set.
Significance. If correct, the result would be the first higher-dimensional shrinking target theorem for beta-dynamical systems, extending the one-dimensional results in a natural direction. The upper bound is clean and the pressure formalism is appropriate. The lower-bound strategy is plausible and follows a known Cantor-set pattern. However, the manuscript is not self-contained in several load-bearing places: the lower bound assumes f >= (1+epsilon)g without a reduction from f >= g, the key measure estimate in Step 3 is delegated to [18] and [22] without stating hypotheses or proving transfer from the beta_N subshift to the beta-system, and the containment F_infty subset E(T_beta, f, g) is asserted without accounting for the difference between S_n f(x*) and S_n f(x). These gaps currently prevent verification of the main theorem, though they appear fixable within the manuscript's scope.
major comments (4)
- [Section 4, beginning] The lower bound proof starts by fixing epsilon > 0 and assuming f(x) >= (1+epsilon)g(y), but Theorem 1.1 only assumes f(x) >= g(y). No reduction from the weaker hypothesis to the separated case is given. This is load-bearing because the construction's choice of k_i >= l_i and the 'shorter side' covering depend on the separation; for f = g the argument as written does not apply. A standard limiting argument with f + delta would need to be supplied explicitly.
- [Section 4.4.1, Case I, Step 3] The essential estimate sum_{U2,W2 in Sigma^h_{beta_N}} ... <= beta^{l epsilon} is asserted by reference to [18, pp. 2095-2097] and [22, pp. 1331-1332] without proof. The displayed formula contains index inconsistencies: the first summation range is written as Sigma^l_beta although the word (epsilon_{l+1},...,epsilon_{m_i}) has length h, the sums switch from Sigma_beta to Sigma_{beta_N} with no explanation, and the factorization of the normalization equation ignores the mandated 0^N blocks. The paper states no hypotheses under which the cited estimate transfers to the present two-dimensional beta-system. Since this estimate is the core of the mass-distribution bound, the lower bound is not established as written. The same issue appears in Case II, Step III.
- [Section 4, construction of F_infty] The paper asserts that F_infty subset E(T_beta, f, g), but the construction only guarantees |T^{n_i} x - x0| < e^{-S_{n_i} f(x*_i)} at the times n_i, where x*_i is a point in the same n_i-cylinder as x. The set E(T_beta, f, g) requires the bound with e^{-S_{n_i} f(x)}. For arbitrary continuous f, S_n f can differ by O(n) for points in the same n-cylinder, so the containment is not automatic. The upper bound handled this with the sandwiching by f +/- delta; the lower bound does not supply an analogous argument.
- [Section 4.4.1, definition of s_i] The normalization equation defining s_i reads sum_{U,W in Sigma^{m_i}_{beta_N}} e^{S f} e^{S g} (1/(beta^{m_i} e^{S f}))^s = 1. For the total mass of children to equal the parent mass, the factor e^{S f} e^{S g} should be e^{S f}/e^{S g}; as written the measure is not a probability measure. This appears to be a typo, but it affects the subsequent estimates and should be corrected explicitly.
minor comments (5)
- [Section 3.1] In the displayed bound for |J_n(W)|, the exponent should be e^{-S_n g(y*)} rather than e^{-S_n f(y*)}.
- [Section 4.4.1, Step 3] The notation (x'_i, y'_i) = (T^l_beta x'_i, T^l_beta y'_i) reuses the same symbols for shifted points, which is confusing; different symbols should be introduced.
- [Section 2, Proposition 2.8] The condition 2n^2 beta < beta^{(n-1)epsilon} is stated without derivation; the proof would be easier to follow if the choice of n0 were explained.
- [Section 4.4.1] The convergence s_i -> s0 is attributed to [18, Theorem 4.1] but the precise statement and the uniformity in the parameters are not given; the reader cannot verify the step without consulting the reference.
- [References] The paper would benefit from citing the specific lemma or theorem in [18] and [22] that is being invoked in Step 3, rather than referencing whole page ranges.
Circularity Check
No significant circularity: the dimension formula is derived from independent pressure-function roots; the proof's main weakness is an unproved transferred estimate, which is a soundness gap, not circularity.
full rationale
The paper's central claim, dim_H E(Tβ,f,g) = min{s1,s2}, is not an input to itself. The numbers s1 and s2 are defined before the proof as infima of pressure conditions, and the upper bound derives them from a direct covering of the rectangles Jn(U) × Jn(W), while the lower bound constructs a Cantor subset and a measure with exponents si defined as finite-scale pressure roots; the convergence si → s0 is imported from the continuity of pressure with respect to β, an external result [18, Theorem 4.1]. No fitted parameter is renamed as a prediction, and no ansatz is smuggled in via a self-citation: the only apparent self-reference in the text appears in a bibliographic list of general β-expansion results ([11]), not as the justification of the main theorem. The load-bearing estimate in §4.4.1, Step 3, bounding the sum over Σ_l^β × Σ_h^β by β^{lε} 'with similar arguments as in [18, pp. 2095-2097] and [22, pp. 1331-1332]' is delegated to prior work and has apparent index inconsistencies (l vs h, β_N vs β), but delegation of a technical estimate is a proof-completeness or transfer issue, not a reduction of the theorem to its own assumptions. Accordingly, the derivation chain is independent of the conclusion and the appropriate circularity score is 0.
Assumptions & free parameters
assumptions (5)
- standard math Parry's admissibility criterion (Lemma 2.2) and Rényi's counting estimate #Sigma_n^beta = exp(n log beta + o(n)) (Lemma 2.3).
- standard math Fan-Wang characterization of full cylinders (Proposition 2.5) and Bugeaud-Wang density of full cylinders among every n+1 consecutive cylinders (Lemma 2.6).
- domain assumption The estimate Sigma_{U2,W2} ... <= beta^{l epsilon} from [18, pp. 2095-2097] and [22, pp. 1331-1332] applies in the present beta-system construction.
- ad hoc to paper The stronger inequality f >= (1+epsilon)g can be assumed without loss in the lower bound.
- standard math Continuity of the pressure function P(T_beta, .) with respect to beta (cited as [18, Theorem 4.1]).
Cite this review
Pith. "Pith review of Higher dimensional shrinking target problem in beta dynamical systems." pith.science (2026). https://pith.science/paper/X37F5F2O
@misc{pith2026190802098,
author = {Pith},
title = {Pith review of: Higher dimensional shrinking target problem in beta dynamical systems},
year = {2026},
howpublished = {\url{https://pith.science/paper/X37F5F2O}},
note = {Machine review of arXiv:1908.02098}
}
abstract
We consider the two dimensional shrinking target problem in the beta dynamical system for general $\beta>1$ and with the general error of approximations. Let $f, g$ be two positive continuous functions. For any $x_0,y_0\in[0,1]$, define the shrinking target set $$ E(T_\beta, f,g):=\left\{(x,y)\in [0,1]^2: \begin{array}{ll} |T_{\beta}^{n}x-x_{0}|<e^{-S_nf(x)}\\ [1ex] |T_{\beta}^{n}y-y_{0}|< e^{-S_ng(y)} \end{array} \ {\text{for infinitely many}} \ n\in \N \right\}, $$ where $S_nf(x)=\sum_{j=0}^{n-1}f(T_\beta^jx)$ is the Birkhoff sum. We calculate the Hausdorff dimension of this set and prove that it is the solution to some pressure function. This represents the first result of this kind for the higher dimensional beta dynamical systems.
Reference graph
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