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arxiv: 2605.30033 · v1 · pith:4Q56D2ZWnew · submitted 2026-05-28 · 🧮 math.CA · math.CO

On hyperbolic corners and unit-area triangles in planar sets of large measure

Aleksandar Bulj , Vjekoslav Kova\v{c} This is my paper

Pith reviewed 2026-06-28 23:52 UTC · model grok-4.3

classification 🧮 math.CA math.CO
keywords hyperbolic cornersunit-area trianglesmeasure boundsErdős conjecturetrilinear smoothing inequalityplanar setsavoiding configurationsRiesz energy
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The pith

Measurable sets in [0,R]² avoiding vertices of axis-aligned right triangles of area 1/2 have measure O(R²/(log R)^c) for any c<1/4.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies large measurable sets in the plane that avoid three points forming an upward right triangle with legs parallel to the axes and area exactly 1/2. It shows that any such set inside the square [0,R]² must have area at most a constant times R² divided by a positive power of log R, where the power can be any number less than 1/4. The argument relies on a new hyperbolic form of a trilinear smoothing inequality to control the density of the set. When the avoided triangles are required only to have some fixed area rather than specifically 1/2, the same method yields a stronger saving of any power less than 1/2. A matching-style construction demonstrates that sets of size roughly R log R can still avoid the configuration, so the upper bound is not sharp but improves earlier o(R²) results and gives partial progress toward a conjecture of Erdős asking whether the measure stays bounded.

Core claim

For large R, any measurable A subset of [0,R]² that contains no triple (x,y), (x+t,y), (x,y+1/t) with t>0 satisfies |A| = O_c(R²/(log R)^c) whenever c<1/4. The same conclusion holds with any c<1/2 when A avoids all triangles of one fixed positive area. The proofs use a hyperbolic variant of the two-dimensional trilinear smoothing inequality together with, in the fixed-area case, induction on scales that alternately controls density and Riesz energy.

What carries the argument

Hyperbolic variant of the two-dimensional trilinear smoothing inequality, which bounds integrals over the hyperbolic corner configurations (x,y), (x+t,y), (x,y+1/t).

If this is right

  • Avoiding the specific area-1/2 configuration forces the measure to be o(R²) with at least a small logarithmic factor.
  • Avoiding all triangles of one fixed area yields a quantitatively stronger logarithmic upper bound.
  • The fixed-area result supplies partial progress on Erdős's question whether the measure must remain O(1).
  • A separate construction shows that measure Ω(R log R) is achievable while avoiding the area-1/2 configuration.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The induction-on-scales technique that controls both density and Riesz energy may transfer to other geometric-configuration problems that mix additive and multiplicative structure.
  • Closing the gap between the proven upper bound O(R² / (log R)^c) and the lower bound Ω(R log R) would require either a stronger smoothing inequality or a different method.
  • If analogous hyperbolic smoothing inequalities exist in higher dimensions, similar measure bounds could apply to corner-avoiding sets in R^d.

Load-bearing premise

The hyperbolic variant of the trilinear smoothing inequality must hold with the quantitative bounds needed to close the density estimates.

What would settle it

An explicit set A inside [0,R]² of measure larger than C R² / (log R)^{1/4} that still contains no triple of the form (x,y), (x+t,y), (x,y+1/t), or a counterexample showing the hyperbolic smoothing inequality fails to deliver the required decay.

Figures

Figures reproduced from arXiv: 2605.30033 by Aleksandar Bulj, Vjekoslav Kova\v{c}.

Figure 1
Figure 1. Figure 1: Example for the lower bound. On the one hand, since a band {(x, y) ∈ [0, R] 2 : a ⩽ x + y ⩽ b} has area (b 2 − a 2 )/2 for any 0 ⩽ a < b ⩽ R, the whole set AR has measure |AR| = 1 2 Xm j=1  R − 4j + 1 8j 2 − (R − 4j) 2  = R 8 Xm j=1 1 j − m 2 + 1 128 Xm j=1 1 j 2 = 1 8 R log R + O(R). On the other hand, AR contains no triple of the form (1.1). Namely, if it did contain such a triple for some x, y ∈ R a… view at source ↗
read the original abstract

For large $R$, we consider measurable sets $A\subseteq [0,R]^2$ that avoid triples of points of the form $(x,y)$, $(x+t,y)$, $(x,y+1/t)$ with $x,y\in\mathbb{R}$ and $t>0$, i.e., the vertices of upward-oriented, axis-aligned right triangles of area $1/2$. We prove that the measures of such sets satisfy $|A|= O_c(R^2/(\log R)^c)$ for any constant $c<1/4$. An ingredient in the proof is a hyperbolic variant of the two-dimensional trilinear smoothing inequality by Christ, Durcik, and Roos. The aforementioned upper bound is complemented with an example of a set of measure $\Omega(R\log R)$ avoiding the same point configuration. Next, we study measurable sets $A\subseteq [0,R]^2$ that avoid triples of points spanning a triangle of a given fixed area and establish a sharpening of the aforementioned upper bound to any $c<1/2$. This makes partial progress on a question by Erd\H{o}s, who conjectured an upper bound $O(1)$, and improves over a quantitatively weak $o(R^2)$ result by Graham. The latter proof additionally uses induction on scales to interchangeably control the density and the Riesz energy of the set $A$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper proves that for large R, measurable sets A ⊆ [0,R]² avoiding the configuration (x,y), (x+t,y), (x,y+1/t) (t>0) satisfy |A| = O_c(R²/(log R)^c) for any c<1/4, complemented by a construction of measure Ω(R log R). For sets avoiding triangles of any fixed area the bound improves to any c<1/2. The proofs rely on a hyperbolic variant of the Christ-Durcik-Roos trilinear smoothing inequality together with an induction-on-scales argument that controls both density and Riesz energy.

Significance. If the hyperbolic smoothing inequality holds with constants permitting the stated iteration, the results give the first polylogarithmic improvements over the trivial bound for the hyperbolic-corner problem and advance Erdős' unit-area triangle question beyond Graham's o(R²) result, reaching nearly R²/(log R)^{1/2} in the fixed-area case.

major comments (1)
  1. [§4–5] The induction-on-scales argument in §4–5 invokes the hyperbolic variant of the Christ-Durcik-Roos trilinear smoothing inequality; the manuscript must establish explicit dependence of the constants on the hyperbolic parameter and on the scale of the tested configuration (x,y),(x+t,y),(x,y+1/t) so that the iteration yields a positive power c<1/4 (or c<1/2) without the smoothing constant deteriorating faster than any fixed power of log R.
minor comments (1)
  1. [abstract] The abstract states the bounds but does not indicate the precise range of t or the hyperbolic angle over which the smoothing inequality is proved; a brief statement of the parameter regime would clarify applicability to the induction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, the positive evaluation of the results, and the specific suggestion regarding constant dependence. We address the major comment below and will incorporate the requested details in the revision.

read point-by-point responses

  1. Referee: [§4–5] The induction-on-scales argument in §4–5 invokes the hyperbolic variant of the Christ-Durcik-Roos trilinear smoothing inequality; the manuscript must establish explicit dependence of the constants on the hyperbolic parameter and on the scale of the tested configuration (x,y),(x+t,y),(x,y+1/t) so that the iteration yields a positive power c<1/4 (or c<1/2) without the smoothing constant deteriorating faster than any fixed power of log R.

    Authors: We agree that explicit control on the constants is required to justify the iteration and obtain the stated exponents. The current manuscript invokes the hyperbolic smoothing inequality (Theorem 3.1) but does not spell out the dependence on the hyperbolic parameter t and the dyadic scale in sufficient detail for the induction. In the revised version we will expand the proof of the smoothing inequality (or add an appendix) to record that the constant is at most C_ε (1 + |log t|)^C (scale)^ε for any ε>0, with C independent of t and the scale. This growth is slow enough that, when fed into the induction-on-scales argument of §§4–5, the accumulated loss remains smaller than any fixed power of log R, yielding the claimed c<1/4 for the area-1/2 case and c<1/2 for the fixed-area case. revision: yes

Circularity Check

0 steps flagged

No circularity; bounds derived from external inequality and induction on scales

full rationale

The paper establishes measure upper bounds for sets avoiding hyperbolic corners or fixed-area triangles by citing and applying a hyperbolic variant of the external Christ-Durcik-Roos trilinear smoothing inequality, then using induction on scales to control density and Riesz energy. No self-definitional reductions, no parameters fitted to data then relabeled as predictions, and no load-bearing self-citations appear in the derivation chain. The cited smoothing result is from independent authors and is treated as an external analytic input rather than derived within the paper. The complementary lower-bound construction is an explicit example, not a tautology. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on standard properties of Lebesgue measure in the plane together with the asserted validity of a new hyperbolic smoothing inequality; no free parameters or invented entities are introduced.

axioms (2)
  • standard math Lebesgue measurability of subsets of R² and the standard properties of two-dimensional Lebesgue measure
    Used to define |A| and the forbidden point configurations throughout.
  • domain assumption Existence and applicability of a hyperbolic variant of the Christ-Durcik-Roos trilinear smoothing inequality
    Explicitly identified in the abstract as a key ingredient of the proof.

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