{"id":"cc59d8ad-1874-4420-88a5-77a3f44ad306","arxiv_id":"2411.19856","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A set E is right-sided weakly porous if and only if d(.,E)^(-α) belongs to the one-sided Muckenhoupt class A1+ for some α>0 and is locally integrable.","lead":"The paper proves that a set in the real line is right-sided weakly porous exactly when some negative power of its distance function is a one-sided A1 Muckenhoupt weight. It also proves that ordinary weak porosity is equivalent to being both left-sided and right-sided weakly porous.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the unproved estimate in Lemma 4.3 is correct and easily supplied, so the central theorem stands.","rationale":"I read the paper in good faith. The central claim is Theorem 4.5: d(·,E)^{-α} ∈ A_1^+ ∩ L^1_loc for some α iff E is right-sided weakly porous. The reader identified the estimate in Lemma 4.3 as the weakest assumption. That estimate is not proved in the text, but it is true and has a short proof: an E-free interval J of length at least 4ε' has its middle part (points at distance ≥ε' from both endpoints) lying entirely outside E(ε'), because the ε'-ball around such a point is contained in J and hence contains no point of E. Only the two endpoint strips, of total length ≤2ε', can belong to E(ε'). Thus |J \\ E(ε')| ≥ |J| − 2ε' ≥ |J|/2. Applying this to the intervals J_i^j produced by the right-sided weak porosity condition, whose lengths are at least γε = 4ε', gives exactly the bound needed for Lemma 4.3. The rest of the proof checks out: in II⇒I, Case 1 uses a direct distance comparison, Case 2 uses Lemma 4.2 with η = 2, and Case 3 uses Lemma 4.3 to obtain a geometric decay of |F(ε)|, from which the integral estimate follows with α chosen small enough that β1^{-α} β2 < 1. In I⇒II, the reduction to closed E is justified by d(·,E) = d(·,Ē), the measure-zero conclusion follows from local integrability, and the porosity constants are chosen uniformly. Minor typos, such as the constant (1−α) in Case 2's displayed computation (which should be 1/(1−α)), do not affect the argument since the constant is absorbed into C(α). No circularity, unsupported citation, or external-consensus risk emerged. The only slightly under-explained step is the one above, and it is easily completed. Therefore I recommend no change to the reader's ACCEPT verdict.","tokens_in":14307,"tokens_out":46432,"duration_ms":350802,"concrete_test":"Re-derive the estimate in Lemma 4.3: for an E-free open interval J of length L ≥ 4ε', show J ∩ E(ε') has measure at most 2ε' by the ball argument, so |J \\ E(ε')| ≥ L/2; this confirms the contraction factor and closes the only missing derivation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's flagged assertion in Lemma 4.3, that |J_i^j \\ E(ε')| ≥ 1/2 |J_i^j| where ε' = γε/4, is not a real gap. Since J_i^j is an E-free open interval with |J_i^j| ≥ γε = 4ε', any x in J_i^j at distance at least ε' from both endpoints has (x−ε', x+ε') ⊂ J_i^j, so d(x,E) ≥ ε'; hence J_i^j ∩ E(ε') is contained in the two endpoint strips of total length at most 2ε' ≤ |J_i^j|/2. The contraction |F(β1 ε)| ≤ β2 |F(ε)| follows with β1 = γ/4 and β2 = 1 − min{3/8, σ/2}, both in (0,1). I find no other load-bearing concerns: the proof of Theorem 4.5 is consistent, the three cases in II⇒I are exhaustive, the use of Lemma 4.2 in Cases 2 and 3 is valid, and the parameter choices (γ small in I⇒II, α small in Case 3) are legitimate. The paper should be accepted as is.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper introduces one-sided weakly porous sets in R and proves a geometric characterization of when distance powers d(·,E)^{-α} are one-sided A1 weights. The main result (Theorem 4.5) states that for a non-empty set E, d(·,E)^{-α} ∈ A_1^+(R) ∩ L^1_loc(R) for some α>0 if and only if E is right-sided weakly porous. The paper also proves (Theorem 3.7) that a set is weakly porous in the sense of Anderson et al. if and only if it is both left- and right-sided weakly porous. The proof of the main theorem is built on a one-sided maximal-hole function and a decay estimate for the level sets of the distance function (Lemma 4.3), and it yields a Hausdorff dimension bound for right-sided weakly porous sets (Corollary 4.4).","tokens_in":14603,"tokens_out":20957,"duration_ms":166702,"significance":"If correct, the main theorem gives a complete answer to the question of which subsets of R have distance powers in the one-sided A1 class, extending the recent two-sided results of Anderson, Lehrbäck, Mudarra, and Vähäkangas. The manuscript is careful and largely self-contained: it supplies detailed estimates for the weight condition, explicit parameter choices in the porosity direction, and a transparent reduction to closed sets. The examples (N0 and the asymmetric sequence E) are instructive and correctly illustrate the difference between one-sided and two-sided porosity. The paper also ships a nontrivial dimensional consequence (Corollary 4.4), which lends additional support to the naturalness of the definition. The main equivalence is derived directly from the definitions, without circular use of the conclusion, and the only external load-bearing input is Lemma 3.6 from [2].","major_comments":[],"minor_comments":[{"comment":"In Lemma 4.3, after the intervals J_i^j are introduced, the estimate |J_i^j \\ E(ε')| ≥ (1/2)|J_i^j| is asserted without proof; it is true because |J_i^j| ≥ γε = 4ε' and E(ε') can meet J_i^j only in the two endpoint strips of total length at most 2ε' ≤ |J_i^j|/2, but this justification should be included explicitly.","section":"Lemma 4.3"},{"comment":"In the computation of the average of d(x,∂I^-)^{-α}, the equality 2/(b-a) ∫_0^{(b-a)/2} u^{-α} du = (1-α)((b-a)/2)^{-α} is incorrect; the correct factor is 1/(1-α). The subsequent inequality remains valid with the corrected constant, so this is a local error in the displayed formula.","section":"Theorem 4.5, Case 2"},{"comment":"The text refers to a 'consistency diagram' that can be obtained from the main results, but no diagram appears in the manuscript; either include the diagram or delete the reference.","section":"Introduction"},{"comment":"The proof begins 'Assume now that E is a closed (σ,γ,+)-w.p. set' without explicitly justifying the reduction from an arbitrary right-sided weakly porous set; the reduction follows from Proposition 3.5(i) and d(·,E)=d(·,\\bar E), and it should be stated.","section":"Theorem 4.5 (II⇒I)"},{"comment":"In the case I ⊂ (x0,∞), the sentence 'E+ satisfies the w.p. condition on I ... and therefore also satisfies the right-sided w.p. condition' uses a local version of the argument in Theorem 3.7; a brief explanation of why Lemma 3.6 applies to the restricted set on the half-line would improve clarity.","section":"Proposition 3.8(a)"}],"recommendation":"minor_revision","confidential_remarks":"The paper is within the scope of the journal and the main theorem is sound. The self-citation [1] is used only for analogy and does not affect the validity of the results. The only concern is the small number of local gaps, which are easily addressed in revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a solid paper. The one-sided weak porosity notion is genuinely new, and Theorem 4.5—d(·,E)^-α ∈ A1+ ∩ L1_loc for some α > 0 iff E is right-sided weakly porous—is the natural one-sided counterpart to the Anderson–Lehrbäck–Mudarra–Vähäkangas result. The proofs are detailed, the constants are explicit, and the consistency diagram in Theorem 3.7 is a nice addition. The one soft spot the reader flagged is real but small. In Lemma 4.3, the bound |J_i \\ E(ε')| ≥ 1/2 |J_i| is asserted without derivation. The stress-test note supplies the missing argument: since |J_i| ≥ γε = 4ε', the ε'-neighborhood of E can meet J_i only in two endpoint strips of total length at most 2ε' ≤ |J_i|/2. That closes the gap cleanly, so the contraction estimate and Corollary 4.4 stand. I also noticed a couple of typos—for instance, the Case 2 estimate in Theorem 4.5 writes the factor (1−α) in a way that could confuse a reader—but nothing that changes the conclusion. I disagree with any suggestion that the self-citation to [1] is a problem; it is used only for analogy, and the borrowed Lemma 3.6 from [2] is explicitly cited and load-bearing only in Theorem 3.7, where it is applied correctly. The examples (N0 right-sided but not left-sided; the cut-off of Z) are helpful and illustrate the anisotropy well. Bottom line: the central theorem is correct, the new definitions are coherent, and the paper deserves a serious referee. I would send it out and expect a minor-revision outcome. The intended audience is harmonic analysts working on Muckenhoupt weights and geometric measure theorists interested in porosity; they will both find this worth reading.","headline":"Solid, correct paper: the new one-sided weak porosity notion earns its keep and Theorem 4.5 is the right one-sided analogue; the only flagged gap is a minor, easily repairable estimate.","tokens_in":604,"tokens_out":2703,"would_cite":true,"duration_ms":37430,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["28A80","28A75","42B37"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a set $E$ is right-sided weakly porous exactly when some negative power of the distance to $E$ is a locally integrable one-sided $A_1^+$ weight.","keywords":["one-sided Muckenhoupt weights","A1 weights","right-sided weakly porous sets","left-sided weakly porous sets","distance functions","maximal hole function","Hausdorff dimension","weak porosity"],"falsifier":"Take $E=\\mathbb{N}_0$ from Example 3.9, pick an interval $I$ whose right half contains a pore of length $2\\rho(I^+)$, and compute $|F(\\gamma\\varepsilon/4)|$ versus $|F(\\varepsilon)|$ directly for small $\\varepsilon$; a single instance where $|F(\\gamma\\varepsilon/4)|$ exceeds $\\beta_2|F(\\varepsilon)|$ would disprove the contraction lemma as stated, while uniform validity would confirm the geometric engine used to prove Theorem 4.5.","tokens_in":14128,"feed_emoji":"📐","tokens_out":9558,"duration_ms":77348,"temperature":0.7,"pith_summary":"The paper establishes a one-to-one correspondence between one-sided Muckenhoupt weights and one-sided porosity on the real line: a nonempty set $E\\subset\\mathbb{R}$ admits some $\\alpha>0$ for which $d(\\cdot,E)^{-\\alpha}$ belongs to the right-sided Muckenhoupt class $A_1^+(\\mathbb{R})$ and is locally integrable if and only if $E$ is right-sided weakly porous. The new geometric condition says that every interval's left half contains disjoint intervals avoiding $E$ whose total length is a fixed proportion of that half, and each such hole is comparable in size to the largest $E$-free interval in the right half. The proof turns this porosity into exponential decay of the measure of the thickened set $\\{x: d(x,E)<\\varepsilon\\}$ near the left side, which yields the $A_1^+$ inequality and, as a corollary, a Hausdorff-dimension estimate for $E$. The paper also shows that being both left- and right-sided weakly porous is equivalent to the two-sided weak porosity already known to characterize ordinary $A_1$ distance weights, so the two theories are consistent. This matters because it gives a purely geometric description of exactly which distance functions make one-sided maximal operators weighted-bounded, and it exposes the one-sided anisotropy hidden in the classical condition.","feed_headline":"Right-sided porosity characterizes one-sided A1 weights","feed_subtitle":"Negative distance powers are A1+ weights exactly for right-sided weakly porous sets.","key_machinery":"The carrying object is the maximal-hole function $\\rho_E(I)$, the radius of the largest open interval centered somewhere in $I$ and contained in $I\\setminus E$, and the new class of $(\\sigma,\\gamma)$-right-sided weakly porous sets built from it. The mechanism that does the heavy lifting is Lemma 4.3, an exponential-neighborhood contraction: for $\\tilde I$ obtained from the largest hole in $I^+$, the measure of $F(\\varepsilon)=E(\\varepsilon)\\cap\\tilde I$ decays by a fixed factor $\\beta_2$ when $\\varepsilon$ is scaled by $\\beta_1=\\gamma/4$. Iterating that contraction supplies the integrability estimate that closes Case 3 of Theorem 4.5 and, through Corollary 4.4, the Hausdorff-dimension bound.","core_discovery":"The central claim is Theorem 4.5: $d(\\cdot,E)^{-\\alpha}\\in A_1^+(\\mathbb{R})\\cap L^1_{\\mathrm{loc}}(\\mathbb{R})$ for some $\\alpha>0$ if and only if the nonempty set $E$ is right-sided weakly porous. In the forward direction, the right-sided porosity of $E$ is shown to control the distribution of points close to $E$: Lemma 4.3 gives a uniform contraction $|F(\\beta_1\\varepsilon)|\\le \\beta_2|F(\\varepsilon)|$, with $F(\\varepsilon)=E(\\varepsilon)\\cap\\tilde I$, and summing the decayed layers produces the $A_1^+$ estimate on $I^-$. In the reverse direction, the one-sided $A_1^+$ inequality is read directly as the required abundance of large holes: points of $I^-\\setminus E$ whose component is shorter than $2\\gamma\\rho(I^+)$ occupy a fraction of $I^-$ that can be made arbitrarily small by choosing $\\gamma$ small, so the complementary long holes cover a fixed proportion of $I^-$. A consistency theorem identifies the two-sided weakly porous sets with the intersection of the left- and right-sided classes, and Corollary 4.4 bounds the Hausdorff dimension of any right-sided weakly porous set by $1-\\log\\beta_2/\\log\\beta_1$.","pith_inferences":["The contraction ratio in Lemma 4.3 likely ties the smallest admissible weight exponent $\\alpha$ in Theorem 4.5 to the dimension gap $1-\\dim_H E$; the paper states the decay exponent $\\log\\beta_2/\\log\\beta_1$ but does not identify which $\\alpha$ is optimal.","The one-sided definitions are coordinate-dependent, but they suggest a directional version in higher dimensions: fixing a direction and requiring pores to accumulate on one side of every ball should characterize directional $A_1^+$ distance weights, a generalization not attempted here.","A direct check on the accumulated set of Example 3.10 would show whether the unproved half-length estimate in Lemma 4.3 is essential or merely an artifact of the proof, since that set is right-sided weakly porous and its neighborhoods can be computed explicitly."],"forward_implications":["A right-sided weakly porous set is exactly the kind of set for which some negative power of the distance is a locally integrable $A_1^+$ weight; every set that produces such a weight must be right-sided weakly porous.","Every right-sided weakly porous set has zero Lebesgue measure and Hausdorff dimension at most $1-\\log\\beta_2/\\log\\beta_1$, with $\\beta_1,\\beta_2$ the contraction constants of Lemma 4.3.","A set is weakly porous in the two-sided sense if and only if it is simultaneously left- and right-sided weakly porous, so the classical $A_1$ distance-weight theorem is the symmetric special case of the one-sided theorem.","By reflection symmetry, the mirror statement holds: $E$ is left-sided weakly porous if and only if $d(\\cdot,E)^{-\\alpha}\\in A_1^-(\\mathbb{R})\\cap L^1_{\\mathrm{loc}}(\\mathbb{R})$ for some $\\alpha>0$.","Right-sided weak porosity is not inherited by arbitrary subsets, as the set $\\mathbb{N}_0$ shows, even though every tail of a two-sided weakly porous set is right-sided weakly porous; the one-sided condition is genuinely anisotropic."],"supporting_citations":[{"why":"defines two-sided weak porosity and establishes the $A_1$ distance-function characterization that Theorem 3.7 aligns with the one-sided theorem.","marker":"[2]"},{"why":"supplies the equivalent $A_1^+$ condition used through Proposition 2.2 and the identity $A_p=A_p^-\\cap A_p^+$.","marker":"[7]"},{"why":"introduces the one-sided Hardy-Littlewood maximal operators and one-sided Muckenhoupt classes that motivate the theorem.","marker":"[10]"},{"why":"provides the weak-porosity formulation in spaces of homogeneous type whose phrasing the authors adapt for the one-sided definitions.","marker":"[1]"},{"why":"gives the upper Minkowski dimension formula used in Corollary 4.4 to bound the Hausdorff dimension of right-sided weakly porous sets.","marker":"[8]"},{"why":"extends weak porosity to metric measure spaces and supplies the generalized context behind the chosen definitions.","marker":"[9]"}],"fun_headline_variants":["Right-weak porosity pinpoints one-sided A1 weights","A1+ weights arise exactly from right-weakly porous sets","New criterion: right-weak porosity for A1+ weights","Right-sided porosity: the geometry behind A1 weights","One-sided Muckenhoupt linked to new porosity concept"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The exponential-contraction estimate for neighborhoods of $E$ rests on the unproved assertion that each pore interval loses no more than half its length when the endpoint strips of width $\\gamma\\varepsilon/4$ are removed; if that assertion fails, the contraction inequality and the main theorem's Case 3 do not follow from the proof given.","fun_headline_variants_meta":{"raw":{"variants":["Right-weak porosity pinpoints one-sided A1 weights","A1+ weights arise exactly from right-weakly porous sets","New criterion: right-weak porosity for A1+ weights","Right-sided porosity: the geometry behind A1 weights","One-sided Muckenhoupt linked to new porosity concept"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000263,"raw_usage":{"total_tokens":1601,"prompt_tokens":945,"completion_tokens":656,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":561,"completion_tokens_details":{"reasoning_tokens":572}},"tokens_in":561,"tokens_out":656,"duration_ms":5888,"temperature":1.0,"reasoning_tokens":572,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T05:47:15.819877+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $E=\\mathbb{N}_0$ from Example 3.9, pick an interval $I$ whose right half contains a pore of length $2\\rho(I^+)$, and compute $|F(\\gamma\\varepsilon/4)|$ versus $|F(\\varepsilon)|$ directly for small $\\varepsilon$; a single instance where $|F(\\gamma\\varepsilon/4)|$ exceeds $\\beta_2|F(\\varepsilon)|$ would disprove the contraction lemma as stated, while uniform validity would confirm the geometric engine used to prove Theorem 4.5.","supporting_citations":[{"cited_title":"Anderson, Juha Lehrb¨ ack, Carlos Mudarra, and Antti V","cited_arxiv_id":null,"evidence_quote":"defines two-sided weak porosity and establishes the $A_1$ distance-function characterization that Theorem 3.7 aligns with the one-sided theorem."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the equivalent $A_1^+$ condition used through Proposition 2.2 and the identity $A_p=A_p^-\\cap A_p^+$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduces the one-sided Hardy-Littlewood maximal operators and one-sided Muckenhoupt classes that motivate the theorem."},{"cited_title":"Geometry of sets and measures in Euclidean spaces , volume 44 of Cambridge Studies in Advanced Mathematics","cited_arxiv_id":null,"evidence_quote":"gives the upper Minkowski dimension formula used in Corollary 4.4 to bound the Hausdorff dimension of right-sided weakly porous sets."},{"cited_title":"Weak porosity on metric measure spaces, 2024","cited_arxiv_id":null,"evidence_quote":"extends weak porosity to metric measure spaces and supplies the generalized context behind the chosen definitions."}],"review_version":1}