{"id":"88afad03-08ca-431f-81e3-6fcc947d6165","arxiv_id":"2508.18228","paper_version":3,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"For Borel sets X, Y in the plane with X not in a line and dim X > 0, some x in X gives dim(pi_x Y) >= min{(dim X + dim Y)/2, dim Y, 1}.","lead":"This paper proves a stronger lower bound on the dimension of radial projections of planar sets: from any Borel set X not contained in a line and with positive dimension, some point x in X sees any other Borel set Y in dimension at least min{(dim X + dim Y)/2, dim Y, 1}. The result reaches the trivial maximum in several regimes, which makes it a real step in the projection theory of fractal sets.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sharp endpoint dim Y ≤ dim X rests on an unstated multi-scale incidence estimate; no proof text is available to verify it.","rationale":"I agree with the reader that the proof is the issue. The abstract's theorem is internally consistent and passes the obvious line-counterexample check, so I do not claim it is false. But the sharp endpoint t≤s is a strong geometric assertion whose only visible support is an unnamed incidence estimate. Since the full text is absent, the verdict stays UNVERDICTED. I recommend no change from the reader's verdict.","tokens_in":915,"tokens_out":17720,"duration_ms":238705,"concrete_test":"Retrieve the full text and locate the incidence/discretization lemma behind the theorem. Check (1) whether it is stated for all Borel sets or only for compact/Frostman sets; (2) whether the bound is exactly min{(s+t)/2,t,1} with no ε-loss; (3) whether the 'not contained in a line' condition is used to rule out the obvious counterexample Y a line and X⊂Y. If the lemma is absent or weaker, the theorem is unsupported. Independent numerical sanity: for Y a self-similar Cantor set on a line (dim t<1) and X a t-dimensional subset of a non-linear curve, estimate sup_x dim_H(π_x Y) at scales down to 2^-15; the sharp endpoint requires the estimate to approach t, whereas an ε-loss incidence proof would leave a detectable deficit.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract theorem has a striking consequence in the regime t=dim_H Y ≤ s=dim_H X: since dim_H(π_x Y) ≤ t for every x, the claimed lower bound min{(s+t)/2, t, 1}=t forces sup_x dim_H(π_x Y)=t. This says that every Borel set X with dimension s≥t that is not contained in a line must contain a point from which the radial projection of Y preserves the full dimension of Y. Equivalently, the 'bad' set B={x: dim_H π_x(Y)<t} cannot contain any such X. Such a strong non-containment cannot be read off from the abstract; in this area it would follow from a multi-scale incidence estimate controlling the number of pairs (x,y)∈X×Y whose connecting directions fall in a narrow angular window. The abstract neither states this estimate nor indicates where it is proved, and the full text is unavailable. If the incidence estimate requires compactness, doubling, or Frostman conditions on X and Y, the reduction from arbitrary Borel sets could fail; if it has a logarithmic loss, the exact exponent (s+t)/2 in the complementary regime would not be obtained. This is the central correctness risk, not a contradiction in the statement itself.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper, as available to the referee, consists solely of an abstract. It claims a universal, parameter-free theorem: for Borel sets X,Y in R^2 with X not contained in a line and dim_H(X)>0, sup_{x in X} dim_H(pi_x Y) >= min{(dim_H(Y)+dim_H(X))/2, dim_H(Y), 1}. The abstract gives no proof, no derivation, and no statement of any supporting lemma or incidence estimate.","tokens_in":1086,"tokens_out":3257,"duration_ms":39393,"significance":"If true, the claimed bound is significant and internally coherent. In the regime dim_H(Y) <= dim_H(X), it yields sup_x dim_H(pi_x Y) = dim_H(Y), saturating the trivial upper bound and asserting that every Borel X of sufficiently large dimension not contained in a line contains a point from which the radial projection of Y preserves the full dimension of Y. In the regime dim_H(X)+dim_H(Y) >= 2, it gives the maximum possible value 1. In the intermediate regime it gives (dim_H(X)+dim_H(Y))/2, which the authors state improves the best known lower bound. The statement is falsifiable and has no free parameters. However, the strength of the middle regime and the sharp endpoint depend on a multi-scale incidence estimate that is completely absent from the submitted text. A mathematical result of this type cannot be assessed without the proof.","major_comments":[{"comment":"The central result is stated without any supporting argument. The manuscript contains no lemmas, no derivation, and no indication of where the proof is located. In this area, a bound of the form (dim_H(Y)+dim_H(X))/2 typically follows from a multi-scale incidence estimate controlling the number of pairs (x,y) in X x Y whose connecting directions lie in a narrow angular window. Neither the estimate nor its proof is stated. This omission is load-bearing: without the incidence estimate, the theorem cannot be verified.","section":"Abstract (theorem statement)"},{"comment":"In the endpoint case, the theorem asserts sup_x dim_H(pi_x Y) = dim_H(Y), meaning full dimension preservation. This is a strong structural statement about every Borel set X with dim_H(X) >= dim_H(Y) that is not contained in a line. The abstract gives no indication of how the reduction from arbitrary Borel sets to compact, doubling, or Frostman-regular sets is performed, nor what quantitative control is needed at every scale. If the approximation step requires additional hypotheses, the universal statement would fail. This needs to be spelled out in the proof.","section":"Abstract (endpoint case dim_H(Y) <= dim_H(X))"}],"minor_comments":[{"comment":"The quantities dim_H and pi_x are not explicitly defined. They are standard in the field, but definitions would make the abstract self-contained.","section":"Abstract (notation)"},{"comment":"The phrase 'improve the best known lower bound' is not accompanied by the previous record or a comparison. The authors should state the prior bound and the regime in which the new bound is strictly better.","section":"Abstract (comparison with prior work)"},{"comment":"The submission contains no references. For a complete paper, references to prior work on radial projections are expected.","section":"General"}],"recommendation":"uncertain","confidential_remarks":"The manuscript as received contains only an abstract. There is no proof text to evaluate. The claimed theorem is plausible and, if true, important, but the absence of the full argument makes any verdict other than 'uncertain' inappropriate. If the full paper exists, it should be provided for review."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper from its abstract alone, because that's all we have. The theorem: for Borel X, Y in the plane, X not in a line, dim X > 0, the supremum over x in X of dim(pi_x Y) is at least min{(dim Y + dim X)/2, dim Y, 1}. That's a nice package: it respects the trivial upper bound, it gives the full dimension when dim Y <= dim X, and it reaches 1 when the dimensions sum to at least 2. If the proof works, it genuinely improves the best known lower bound, and the endpoint behavior is exactly what you'd hope for. No free parameters, no fitted constants, just a clean universal statement. Full credit for the formulation. The soft spot is obvious: there is no proof text. The stress-test note is right that the dim Y <= dim X case is the sharpest part - it says that any set X with dimension at least dim Y must contain a point from which the radial projection of Y preserves full dimension. That's a strong conclusion. It would follow from a multi-scale incidence estimate controlling pairs (x,y) whose connecting directions are clustered. The abstract doesn't state such an estimate, and we can't see whether it has logarithmic losses or requires extra assumptions like compactness or Frostman conditions. Those are the things a referee has to check. But that's not a flaw in the abstract; it's just an unverifiable claim until the full text appears. The authors are established in this area, and the statement is coherent on its own terms. I don't see any sign of circularity or fitting. The main risk is that the incidence estimate doesn't deliver the exact exponent (s+t)/2 in the middle regime, or the reduction from Borel to nice sets quietly restricts the conclusion. Those are standard worries for this type of argument, and the only way to resolve them is to read the proof. Who is this for? Geometric measure theorists working on radial projections, Furstenberg sets, or dimension conservation. It deserves a serious referee, not a desk reject. If I were handling it, I'd send it out to someone who knows the Liu-type bounds cold and ask specifically about the multi-scale incidence step and the Borel reduction. My own verdict is unverified, but the claim is significant enough to spend referee time on.","headline":"Abstract-only look at a clean, sharp radial projection bound; the strong endpoint regime is exactly where the proof needs the most scrutiny.","tokens_in":630,"tokens_out":691,"would_cite":false,"duration_ms":28406,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["28A78","28A80"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any two Borel sets in R², some radial projection has dimension at least (dim X + dim Y)/2 unless a simpler bound already applies.","keywords":["Hausdorff dimension","radial projections","planar Borel sets","dimension distortion","incidence estimates","fractal geometry"],"falsifier":"Construct two Borel sets X,Y ⊂ R², with X not contained in a line and dim_H(X) = s, dim_H(Y) = t satisfying t > s and t + s < 2, for which sup_{x∈X} dim_H(π_x Y) < (t + s)/2. Such a pair would directly contradict the theorem.","tokens_in":720,"feed_emoji":"📐","tokens_out":6910,"duration_ms":86804,"temperature":0.7,"pith_summary":"This paper establishes a new lower bound on how large radial projections of planar sets must be. For any two Borel sets X and Y in R², with X not lying on a line and having positive Hausdorff dimension, there exists a point x in X from which the radial projection of Y has Hausdorff dimension at least min{(dim Y + dim X)/2, dim Y, 1}. The first two terms mean that whenever dim Y ≤ dim X, the bound is exactly dim Y, the largest possible; when the two dimensions sum to at least two, it is the full circle dimension 1. The genuinely new part is the average (dim Y + dim X)/2 in the remaining case, which improves the previous best known lower bound. If the result is correct, it identifies the exact dimension profile of the problem except possibly in one open triangle of dimension pairs.","feed_headline":"New floor for radial projections in the plane","feed_subtitle":"Two Borel sets force a projection of at least the average Hausdorff dimension in the hard cases.","key_machinery":"The central object is the radial projection map π_x: Y → S¹, which sends each y to the direction of the ray from x through y. The theorem is a statement about the entire family of these maps indexed by centers x∈X. The minimum formula reflects the trivial upper bound dim_H(π_x Y) ≤ min{dim_H Y, 1}, so the proof's task is to show that, except in the hard triangle, this ceiling is reached, and in the hard triangle that the average is a floor. The abstract does not display the underlying incidence estimate, but the sum-of-dimensions in the bound is characteristic of a discretized multi-scale counting argument that controls how many pairs (x,y) produce directions lying in a narrow angular cone.","core_discovery":"The paper's theorem, stated in the abstract, is that for Borel sets X,Y ⊂ R² with X not contained in any line and dim_H(X) > 0, sup_{x∈X} dim_H(π_x Y) ≥ min{(dim_H(Y) + dim_H(X))/2, dim_H(Y), 1}. In words: looking at all radial projections of Y from centers in X, at least one of them has Hausdorff dimension as large as the minimum of the two individual dimensions, the average of the two, and 1—whichever is smallest. Since dim_H(π_x Y) ≤ min{dim_H(Y), 1} is the trivial upper bound, the theorem says this ceiling is actually attained whenever dim_H(Y) ≤ dim_H(X), and also whenever the two dimensions sum to at least 2. The only region where the guaranteed value falls below the upper bound is the","pith_inferences":["The formula's sharpness in the dim_H(Y) ≤ dim_H(X) regime hints that the true supremum might actually be min{dim_H(Y), 1} for all X with positive dimension, and the average in the hard triangle could be an artifact of the current incidence estimate.","A natural testable extension is whether the same phase-transition formula carries over to radial projections in R^n onto S^{n−1}, with the 1 replaced by the sphere dimension n−1.","The assumption dim_H(X) > 0 appears necessary: a zero-dimensional set of centers likely cannot force any projection to be large, so the threshold at dim_H(X) = 0 is a sharp corner."],"forward_implications":["Whenever dim_H(Y) ≤ dim_H(X), the theorem forces sup_x dim_H(π_x Y) = dim_H(Y), fully closing the problem in that regime and matching the trivial upper bound.","Whenever dim_H(X) + dim_H(Y) ≥ 2, some radial projection from X has full dimension 1, meaning the directions from some center form a dimension-1 subset of the circle.","The only dimension pairs not already settled at the upper bound are those with dim_H(X) < dim_H(Y) and dim_H(X) + dim_H(Y) < 2; there the guarantee is exactly the arithmetic mean of the two dimensions.","The bound is monotone in both dimensions, so passing to larger Borel supersets cannot reduce the guaranteed projection dimension."],"supporting_citations":[],"fun_headline_variants":["Radial projections hit average dimension bound","New floor for planar radial projections","Borel sets force large radial projections","Plane radial projections: average dimension bound","Radial projection size bounded by dimension average"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The proof's load-bearing premise is a multi-scale incidence estimate controlling how many pairs (x,y) produce directions inside a narrow angular cone, and the abstract does not state or locate that estimate; if it fails at some scale, the average bound (dim X + dim Y)/2 does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Radial projections hit average dimension bound","New floor for planar radial projections","Borel sets force large radial projections","Plane radial projections: average dimension bound","Radial projection size bounded by dimension average"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000687,"raw_usage":{"total_tokens":2913,"prompt_tokens":665,"completion_tokens":2248,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":409,"completion_tokens_details":{"reasoning_tokens":2186}},"tokens_in":409,"tokens_out":2248,"duration_ms":21456,"temperature":1.0,"reasoning_tokens":2186,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T16:32:21.455602+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct two Borel sets X,Y ⊂ R², with X not contained in a line and dim_H(X) = s, dim_H(Y) = t satisfying t > s and t + s < 2, for which sup_{x∈X} dim_H(π_x Y) < (t + s)/2. Such a pair would directly contradict the theorem.","supporting_citations":[],"review_version":1}