{"id":"88dc0b16-9513-41d0-a548-b86328a80a76","arxiv_id":"2411.16045","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For beta-dynamical systems with integer bases, the Hausdorff f-measure of shrinking target sets is zero or full according to divergence of an explicit series.","lead":"This paper proves a zero-full dichotomy for Hausdorff measures of shrinking target sets in beta-dynamical systems. It answers a gap that was open even in one dimension and gives a more complete measure-theoretic picture than earlier dimension results.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The divergence proof of Theorem 1.2 restricts to indices with s_n ∏β_i^n ≤ 1; admissible examples can have every term above 1, so the proof does not cover the stated dichotomy.","rationale":"The reader's weakest assumption identifies exactly the load-bearing gap in the paper. The divergence part of Theorem 1.2 is the central new claim, and its proof rests on the reduction to the index set P in equation (4.8), where terms are bounded above by 1. The proof then constructs a lim-sup set of balls of full Lebesgue measure and applies Theorem 2.3. If P is empty, no such balls are constructed. The numerical example with d=2, β_1=β_2=2, f(r)=r^{1/2}, ψ_i(n)=2^{-n/2} satisfies all hypotheses of Theorem 1.2 and has every term s_n ∏ β_i^n equal to 2^{5n/4}, so the claimed pigeonhole reduction cannot be valid in general. This is not merely a technical nuisance: the conclusion of the theorem for that example asserts full H^f-measure, but the proof as written supplies no argument for it. For d=1 the later proof of Theorem 1.4(1) may provide a repair, but no such repair is given for d≥2, and the text does not invoke it in the proof of Theorem 1.2. I therefore agree with the reader's rejection, while noting that the theorem may well be true and the gap may be repairable by handling the tail {n: s_n ∏ β_i^n > 1} separately.","tokens_in":29566,"tokens_out":6812,"duration_ms":62324,"concrete_test":"Run the divergence proof of Section 4.3 on the explicit admissible data d=2, β_1=β_2=2, f(r)=r^{1/2}, ψ_1(n)=ψ_2(n)=2^{-n/2}. Compute s_n and observe that s_n β_1^n β_2^n = 2^{5n/4} > 1 for all n, so the set P defined in (4.8) is empty. Then check whether any part of Section 4.3 can be executed without a single n ∈ P; if no alternative treatment of large terms is present, the divergence part of Theorem 1.2 is not proved for this case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation (4.8) defines P as the set where n^{-2} ≤ s_n(Ψ,f) ∏_{i=1}^d β_i^n ≤ 1, and the proof then constructs the entire divergence argument, including ω_n, the balls B(y,ω_n), and the mass distribution measure, only for n ∈ P. The paper claims this restriction follows by pigeonhole, but that is not valid unless the removed large terms are shown to be harmless. Large terms can occur under the hypotheses of Theorem 1.2. For a concrete admissible instance, take d=2, β_1=β_2=2, f(r)=r^{1/2}, and ψ_1(n)=ψ_2(n)=2^{-n/2}. Then f ≺ d, and for d=2 the condition 'either 1 ⪯ f or f ⪯ 1' holds with f ≺ 1. For τ=2^{-n}, the term in s_n is f(2^{-n})=2^{-n/2}; for τ=2^{-3n/2}, the term is f(2^{-3n/2})=2^{-3n/4}. Hence s_n = 2^{-3n/4}, and s_n β_1^n β_2^n = 2^{5n/4}, which exceeds 1 for every n. The series in Theorem 1.2 diverges, but P is empty, so the construction in Section 4.3 produces no covering balls and the mass distribution proof cannot begin. The same obstruction occurs in the one-dimensional example d=1, β=2, f(r)=r^{1/2}, ψ(n)=2^{-n/2}, where s_n β^n = 2^{n/4}; Section 6 eventually proves the one-dimensional case through Theorem 1.4(1), but that is not the argument supplied for Theorem 1.2. Additionally, the proof uses the unstated assumption ψ_i(n) ≤ 1 in Lemma 4.3(3), although the hypotheses only require ψ_i : N → R_+. The central divergence dichotomy is therefore not derived for the full statement as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Hausdorff f-measure of shrinking target sets associated with beta-transformations on [0,1]^d. Two types of target sets are considered: W_d(Ψ,h), where each coordinate of the orbit must simultaneously fall into a shrinking interval (Theorem 1.2, integer beta), and W_d^×(ψ,h), where the product of the coordinate errors is small (Theorem 1.4, general beta). The main claim is a zero-full dichotomy governed by the convergence or divergence of the series ∑ s_n(Ψ,f)∏_{i=1}^d β_i^n for W_d, and ∑ β_d^{dn} ψ(n)^{-d+1} f(β_d^{-n}ψ(n)) for W_d^×. Theorem 1.3 specializes Theorem 1.2 to the example W_2^*(t). The paper proves the convergence parts cleanly, gives a detailed Chung-Erdős argument for Theorem 1.4(1), and uses a mass-transference framework from the author's earlier work [12] for the divergence parts.","tokens_in":30009,"tokens_out":12797,"duration_ms":114256,"significance":"If correct, these dichotomy laws would give the first complete Hausdorff measure description for these beta-dynamical shrinking target sets, going beyond existing Lebesgue and Hausdorff dimension results. The statements are explicit and contain no fitted parameters, and the one-dimensional product-type dichotomy (Theorem 1.4(1)) appears to be proved rigorously, including for non-integer beta. The convergence part of Theorem 1.2 is also solid. However, the divergence proof of Theorem 1.2 contains a serious gap concerning the restriction to indices with s_n∏β_i^n ≤ 1 and an unstated assumption ψ_i(n) ≤ 1, so the central vector-valued dichotomy is not established as written. The claimed completeness is therefore premature, although the underlying results may be salvageable with a substantial revision.","major_comments":[{"comment":"Equation (4.8) defines P = {n : n^{-2} ≤ s_n(Ψ,f) ∏β_i^n ≤ 1}, and the following proof asserts by pigeonhole that a divergent subseries indexed by P exists. This assertion is false: terms with s_n ∏β_i^n > 1 can occur for every n under the hypotheses of Theorem 1.2. For example, take d=2, β1=β2=2, f(r)=r^{1/2}, and ψ1(n)=ψ2(n)=2^{-n/2}. Then f≺1 and f≺2, so the hypotheses hold. Computing the minimum in the definition of s_n gives s_n = 2^{-3n/4}, attained at τ=2^{-3n/2}, hence s_n ∏β_i^n = 2^{-3n/4} · 2^{2n} = 2^{5n/4} > 1 for all n. The series diverges but P is empty, so the construction of ω_n, the balls B(y,ω_n), and the measure μ in (4.18) cannot start. The one-dimensional analogue d=1, β=2, f(r)=r^{1/2}, ψ(n)=2^{-n/2} has the same obstruction (terms equal 2^{n/4}); that case is covered by Theorem 1.4(1), but the proof of Theorem 1.2 does not cover it, and no alternative argument is supplied for d≥2. This is a load-bearing gap in the divergence half of Theorem 1.2.","section":"§4.3, Eq. (4.8)"},{"comment":"Lemma 4.3(3) asserts that ω_n ≥ β_ℓ^{-n} for k_j+1 ≤ ℓ ≤ d. Its proof uses the inequality ψ_i(n) ≤ 1 for all 1 ≤ i ≤ d, explicitly stated as 'where we have used ψ_i(n) ≤ 1 for all 1 ≤ i ≤ d.' The hypotheses of Theorem 1.2 only require ψ_i : N → R_+ and do not impose ψ_i(n) ≤ 1. If some ψ_i(n) > 1, the factor ∏_{i=1}^m ψ_i(n)^{-1} in the definition of ω_n is smaller than 1, and the lower bound ω_n ≥ β_{m+1}^{-n} n^{-2} > β_ℓ^{-n} need not follow from s_n ∏β_i^n ≥ n^{-2}. Thus the geometric property (3), which is used in the covering and mass distribution argument, is not established under the stated assumptions. The proof needs either an explicit truncation argument showing that one may assume ψ_i(n) ≤ 1 without loss of generality, or a modification of the construction.","section":"§4.3, Lemma 4.3(3)"}],"minor_comments":[{"comment":"Theorem 4.1 is stated for d-tuples of positive functions ψ_i : N → R_+, but the divergence proof defines Φ with ϕ_i(n) = 0 for n ∉ P. As written, the statement 'by Theorem 4.1 the set W_d(Φ,h) is of full Lebesgue measure' is not a valid application of the theorem unless it is extended to nonnegative target functions; this can likely be repaired by taking small positive values for n ∉ P.","section":"§4.3, application of Theorem 4.1"},{"comment":"In the proof of Lemma 2.12, the assertion that appending zeros to a non-full block yields a full cylinder of length β^{-n} is used without proof; a short justification using the definition of full cylinders and the properties of concatenation would improve the clarity.","section":"§2.2, Lemma 2.12"},{"comment":"Theorem 1.3 is derived from Theorem 1.2 in the cases f≺1 of item (1) and 1⪯f of item (2); the revision should explicitly state which parts of Theorem 1.3 depend on a corrected divergence proof of Theorem 1.2.","section":"§5, Theorem 1.3"},{"comment":"There are several typographical errors, including 'generalisity' (§4.3), 'Lebegue' (§4), 'refered' (§1), 'dimenison' (§5), and 'auxilliary' (§6); the notation for Hausdorff measure is also occasionally inconsistent (Hf vs. H^f).","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The central theorem of the paper is not proved as stated: the divergence proof fails for a nonempty class of admissible parameters, including explicit choices with s_n∏β_i^n > 1. The one-dimensional Theorem 1.4(1) is a solid contribution, and the convergence parts are careful. I would be willing to reconsider a revised version that contains a correct treatment of the large-term regime or a suitably restricted theorem statement."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take on He's paper. The problem is real and the stated dichotomy is attractive: a zero-full law for Hausdorff f-measure of shrinking target sets in beta-dynamical systems, which was indeed open even in one dimension. The convergence half of Theorem 1.2 is solid, and the one-dimensional result (Theorem 1.4(1)) is a genuine contribution with a careful proof via the auxiliary set and mass transference. The paper also does a good job situating itself relative to Wang-Wu and the author's own prior framework; the self-citation is legitimate since the prior results are independent.\n\nThe trouble is the divergence half of Theorem 1.2 in the f ≺ d regime. The argument reduces to a set P defined by s_n ∏β_i^n ≤ 1 (Eq. 4.8) and claims a divergent subseries exists on P by pigeonhole. That is not automatic. For d=2, β1=β2=2, f(r)=r^{1/2}, ψ_i(n)=2^{-n/2}, every term s_n β_1^n β_2^n equals 2^{5n/4}, which is >1 for all n. The hypotheses of the theorem are satisfied, the series diverges, but P is empty, so the construction of the balls B(y,ω_n) and the mass distribution never gets started. The same obstruction already appears in d=1. The paper does prove the one-dimensional case separately using a different argument, so the theorem may still be true, but the proof as written does not cover the full statement of Theorem 1.2. There is also an unflagged assumption ψ_i(n) ≤ 1 in Lemma 4.3(3); that is a smaller issue.\n\nSo the central dichotomy is not fully derived as written. That is a load-bearing gap, not a style complaint. The convergence part and the d=1 case are valuable enough that the paper deserves a serious referee, but I would not accept the current version. The author should either restrict the divergence claim to cases where the reduction to P is valid, or supply a separate argument for large terms. If the gap closes, this will be a nice paper.\n\nWho should read it: people working on shrinking targets, beta-expansions, and mass transference principles. I'd bring it to a reading group for the instructive gap, but I wouldn't cite Theorem 1.2 in its current form. Recommendation: send to peer review with a request for major revision; the referee should focus on Section 4.3.","headline":"A well-motivated zero-full dichotomy for Hausdorff measures of shrinking targets in beta-dynamical systems, but the divergence proof of the main theorem has a genuine gap in the f ≺ d case.","tokens_in":30552,"tokens_out":4735,"would_cite":false,"duration_ms":38327,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11J83","11K60"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims a zero-full dichotomy for Hausdorff measures of shrinking target sets in $\\beta$-dynamical systems: $H^f$ is either $0$ or $H^f([0,1]^d)$, according to convergence or divergence of a single explicit series.","keywords":["shrinking target sets","beta-dynamical systems","Hausdorff measure","zero-full dichotomy","Diophantine approximation","full cylinders","mass distribution principle"],"falsifier":"Test the reduction on $d=1$, $\\beta=2$, $f(r)=\\sqrt{r}$, $\\psi(n)=2^{-n/2}$: here $s_n(\\Psi,f)\\prod_{i=1}^d \\beta_i^n = 2^{n/4}$, which is unbounded, so no subseries of terms bounded by $1$ can be extracted. The theorem would still predict $H^{\\sqrt{\\cdot}}(W_1)=H^{\\sqrt{\\cdot}}([0,1])$, so a direct computation of this one-dimensional Hausdorff measure would show whether the divergence argument can be repaired or whether the full statement fails.","tokens_in":29304,"feed_emoji":"🎯","tokens_out":11868,"duration_ms":111279,"temperature":0.7,"pith_summary":"The paper aims to settle, for shrinking target sets in $\\beta$-dynamical systems, the analogue of the classical zero-full law from Diophantine approximation: depending on whether a certain series converges or diverges, the Hausdorff $f$-measure of the set is either zero or equal to the full measure of the unit cube. Such sets are the dynamical versions of points whose orbits hit prescribed shrinking targets infinitely often, and they include both simultaneous and multiplicative variants. While Lebesgue measure and Hausdorff dimension for these sets were already known, their Hausdorff $f$-measure was open even in one dimension; a positive result gives the first complete measure-theoretic description at finer scales. The main theorem treats the simultaneous version when all $\\beta_i$ are integers, and a companion theorem treats the multiplicative version for arbitrary $\\beta>1$ under a restricted range of dimension functions.","feed_headline":"Beta-map shrinking targets have Hausdorff measure zero or full","feed_subtitle":"A single diverging series decides whether the approximable orbits fill the whole cube.","key_machinery":"The carrying object is the layer-cost function $s_n(\\Psi,f)$: for each time $n$ it computes the minimal $f$-cost of covering the rectangles in which the orbit hits its shrinking targets, with the minimum taken over candidate scales $\\tau$ in $\\{\\beta_i^{-n},\\beta_i^{-n}\\psi_i(n)\\}$. The divergence half runs through the exact-length structure of $\\beta$-cylinders: when each $\\beta_i$ is an integer, every level-$n$ cylinder has length exactly $\\beta_i^{-n}$, so target sets contain inscribed balls and can be replaced, up to constants, by limsup sets of balls. A geometric box-splitting argument organizes those balls into separated hyperrectangles, and a carefully built probability measure on them satisfies the mass distribution bound $\\mu(B(x,r))\\ll f(r)/\\omega_n^d$; this verifies a positive Hausdorff $f$-content condition on every ball, which in turn forces the full-measure conclusion. For the multiplicative theorem the analogous role is played by a covering lemma for hyperboloids, by the one-dimensional dichotomy for $W_1(\\psi,h)$, and by a slicing lemma that lifts one-dimensional full measure to higher dimensions.","core_discovery":"On its own terms, the central discovery is Theorem 1.2: for a $d$-tuple of integer $\\beta_i>1$, a dimension function $f$ with $f\\preceq d$ and with $f$ comparable to each intermediate power $r^k$, the Hausdorff $f$-measure of $W_d(\\Psi,h)$ is zero when the series $\\sum_{n=1}^\\infty s_n(\\Psi,f)\\prod_{i=1}^d \\beta_i^n$ converges, and is $H^f([0,1]^d)$ when the series diverges. The quantity $s_n(\\Psi,f)$ is the minimal $f$-cost of covering the $n$th layer of approximating rectangles by balls of one radius $\\tau$, optimized over the candidate radii $\\beta_i^{-n}$ and $\\beta_i^{-n}\\psi_i(n)$. The complementary Theorem 1.4 gives the same zero-full dichotomy for the multiplicative set $W_d^\\times(\\psi,h)$, including the one-dimensional base case $W_1(\\psi,h)$ for arbitrary $\\beta>1$; in dimension $d\\geq 2$ this applies when $(d-1)\\prec f\\preceq s$ for some $s\\in(d-1,d)$. Theorem 1.3 works out the dichotomy explicitly for the known two-dimensional example $W_2^*(t)$, producing a complete Hausdorff-measure description that earlier dimension results did not give.","pith_inferences":["Beyond the paper: the divergence proof uses full cylinders, so the only formal obstruction to extending Theorem 1.2 to non-integer $\\beta_i$ is the missing exact-length property; a testable question is whether a weighted full-cylinder argument recovers the same dichotomy.","Beyond the paper: the convergence half of Theorem 1.2 already holds without integrality, so the zero-full law is plausible in greater generality and may be governed by the same $s_n$ series whenever enough near-full cylinders exist.","Beyond the paper: the explicit reduction to a single series turns the zero-full statement into a computational criterion, so for concrete $\\psi_i$ and $f$ one can classify a shrinking-target set by evaluating the series rather than by constructing covers by hand."],"forward_implications":["If the theorem is right, one series computation decides the fine fractal size of $W_d(\\Psi,h)$: convergence gives $H^f=0$ and divergence gives $H^f=H^f([0,1]^d)$, with no intermediate possibilities.","In the one-dimensional multiplicative case, the dichotomy holds for every $\\beta>1$, giving a Hausdorff-measure zero-full law for $W_1(\\psi,h)$ without an integrality assumption on $\\beta$.","For the explicit example $W_2^*(t)$, Theorem 1.3 determines $H^f$ for dimension functions in the stated comparability classes, including cases such as $f(r)=r^s/\\sqrt{|\\log r|}$ at the critical dimension, where earlier results only identified the Hausdorff dimension.","Whenever the series diverges and $f\\prec d$, the set $W_d(\\Psi,h)$ lies in the class of $G_\\delta$ sets with positive $f$-content in every ball; since that class is closed under countable intersections, all countable intersections of such full-measure sets remain full."],"supporting_citations":[{"why":"Supplies the class $G^f([0,1]^d)$ of $G_\\delta$ sets with positive $f$-content in every ball; countable intersections of such sets have full Hausdorff $f$-measure, which the divergence argument verifies.","marker":"[12]"},{"why":"Provides the Lebesgue zero-full law for limsup sets defined by rectangles used as Theorem 4.1 to build full-measure supersets in the divergence proof.","marker":"[23]"},{"why":"Gives the mass distribution principle used to turn a measure estimate on balls into a lower bound for Hausdorff $f$-content.","marker":"[8]"},{"why":"Rényi's cardinality bounds for admissible sequences, $\\#\\Sigma_\\beta^n\\asymp\\beta^n$, control the number of approximating balls and cylinders.","marker":"[32]"},{"why":"Shows that concatenation of full cylinders is full, the exact-length property needed to inscribe balls in target sets for integer $\\beta$.","marker":"[11]"},{"why":"Supplies lower bounds for the number of full cylinders, used in the one-dimensional auxiliary limsup set and its correlation estimates.","marker":"[30]"},{"why":"Gives the covering of hyperboloid sets by balls with controlled $s$-volume used in the convergence part of the multiplicative theorem.","marker":"[20]"},{"why":"Provides the previous treatment of shrinking targets whose methods are combined with the new geometric argument in the convergence part.","marker":"[28]"},{"why":"Supplies the classical one-dimensional zero-one Lebesgue law for $W_1(\\psi,h)$, the base case for the multiplicative dichotomy.","marker":"[31]"}],"fun_headline_variants":["Zero or full: Hausdorff measure dichotomy in beta-dynamical systems","A series decides: Hausdorff measure of beta-map shrinking targets","Dichotomy resolved: Hausdorff measure of shrinking targets in beta-systems","Beta-dynamical shrinking targets: Hausdorff measure all or nothing","Shrinking targets in beta-systems: Hausdorff measure zero or full"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The divergence half rests on the pigeonhole claim that, whenever the defining series diverges, there is still a divergent subseries whose terms lie between $n^{-2}$ and $1$; if terms larger than $1$ are unavoidable, that reduction may fail.","fun_headline_variants_meta":{"raw":{"variants":["Zero or full: Hausdorff measure dichotomy in beta-dynamical systems","A series decides: Hausdorff measure of beta-map shrinking targets","Dichotomy resolved: Hausdorff measure of shrinking targets in beta-systems","Beta-dynamical shrinking targets: Hausdorff measure all or nothing","Shrinking targets in beta-systems: Hausdorff measure zero or full"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000723,"raw_usage":{"total_tokens":3240,"prompt_tokens":939,"completion_tokens":2301,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":555,"completion_tokens_details":{"reasoning_tokens":2201}},"tokens_in":555,"tokens_out":2301,"duration_ms":13296,"temperature":1.0,"reasoning_tokens":2201,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:37:33.986642+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test the reduction on $d=1$, $\\beta=2$, $f(r)=\\sqrt{r}$, $\\psi(n)=2^{-n/2}$: here $s_n(\\Psi,f)\\prod_{i=1}^d \\beta_i^n = 2^{n/4}$, which is unbounded, so no subseries of terms bounded by $1$ can be extracted. The theorem would still predict $H^{\\sqrt{\\cdot}}(W_1)=H^{\\sqrt{\\cdot}}([0,1])$, so a direct computation of this one-dimensional Hausdorff measure would show whether the divergence argument can be repaired or whether the full statement fails.","supporting_citations":[{"cited_title":"A unified approach to mass transference principle and large intersection property","cited_arxiv_id":"2402.00513","evidence_quote":"Supplies the class $G^f([0,1]^d)$ of $G_\\delta$ sets with positive $f$-content in every ball; countable intersections of such sets have full Hausdorff $f$-measure, which the divergence argument verifies."},{"cited_title":"Kleinbock and B","cited_arxiv_id":null,"evidence_quote":"Provides the Lebesgue zero-full law for limsup sets defined by rectangles used as Theorem 4.1 to build full-measure supersets in the divergence proof."},{"cited_title":"Bishop and Y","cited_arxiv_id":null,"evidence_quote":"Gives the mass distribution principle used to turn a measure estimate on balls into a lower bound for Hausdorff $f$-content."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Rényi's cardinality bounds for admissible sequences, $\\#\\Sigma_\\beta^n\\asymp\\beta^n$, control the number of approximating balls and cylinders."},{"cited_title":"Fan and B","cited_arxiv_id":null,"evidence_quote":"Shows that concatenation of full cylinders is full, the exact-length property needed to inscribe balls in target sets for integer $\\beta$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies lower bounds for the number of full cylinders, used in the one-dimensional auxiliary limsup set and its correlation estimates."},{"cited_title":"Hussain and D","cited_arxiv_id":null,"evidence_quote":"Gives the covering of hyperboloid sets by balls with controlled $s$-volume used in the convergence part of the multiplicative theorem."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the previous treatment of shrinking targets whose methods are combined with the new geometric argument in the convergence part."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the classical one-dimensional zero-one Lebesgue law for $W_1(\\psi,h)$, the base case for the multiplicative dichotomy."}],"review_version":1}