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Rectangles, triangles and Schrödinger waves

This paper employs basic analytic number theory and finite-field constructions to demonstrate how a finite set of lattice points can form many rectangles but few isosceles triangles, thereby providing explicit combinatorial counterexamples to the paraboloid case of the Mizohata–Takeuchi conjecture regarding Schrödinger equation estimates.

Original authors: Jonathan Bennett, Vjekoslav Kovač, Shohei Nakamura, Itamar Oliveira

Published 2026-06-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jonathan Bennett, Vjekoslav Kovač, Shohei Nakamura, Itamar Oliveira

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Game of Shapes and Waves

Imagine you are a detective trying to solve a mystery involving two different worlds:

The Grid World: A giant checkerboard where you place dots (points). You are counting how many perfect rectangles and isosceles triangles (triangles with two equal sides) you can form using these dots.
The Wave World: A physics problem involving the Schrödinger equation (which describes how quantum particles, like electrons, move as waves). Specifically, the paper looks at how "heavy" or "intense" these waves get in certain areas over time.

The authors discovered a surprising secret: The way dots form shapes on a grid directly controls how wild the waves can get. If you can find a grid of dots that makes way more rectangles than triangles, you can prove that certain rules about wave behavior are actually broken.

Part 1: The Shape Puzzle (Combinatorial Geometry)

The Question:
Imagine you have a bag of Lego bricks (dots) on a table. You want to build as many rectangles as possible. But there's a catch: you also have to build triangles. The question was: Is it possible to build a pile of bricks where you have a massive number of rectangles, but very few triangles?

For a long time, mathematicians thought the number of rectangles was always "tamed" by the number of triangles. They thought if you had a lot of rectangles, you must have a lot of triangles to balance the equation.

The Discovery:
The authors (with help from an AI) found a clever way to arrange the dots so that rectangles explode in number while triangles stay small.

The Analogy: Think of a dance floor. Usually, if you have many couples dancing in a square formation (rectangles), you also have many people forming triangles. But the authors found a specific, tricky pattern of dancers where almost everyone is part of a square, but almost no one is part of a triangle.
The Result: They proved that the ratio of rectangles to triangles can get as huge as you want. It's not just a little bigger; it can be astronomically larger.

Part 2: The Wave Problem (Schrödinger Equations)

The Context:
In physics, the Schrödinger equation predicts how a wave moves. Mathematicians have a set of "safety rules" (called Mizohata–Takeuchi estimates) that are supposed to guarantee the wave doesn't get too crazy or "heavy" in any specific spot. These rules act like a speed limit for the wave's intensity.

The Connection:
The paper shows that the "safety rules" for the waves are mathematically linked to the "shape puzzle" from Part 1.

The Logic: If you can arrange your dots to have too many rectangles and too few triangles, it proves that the "safety rules" for the waves are false.
The Metaphor: Imagine the wave rules are a bridge that is supposed to hold a certain weight. The "rectangle-to-triangle" ratio is like a stress test. If the ratio gets too high (too many rectangles), it's like putting too much weight on one side of the bridge. The authors showed that the bridge does break under this specific stress.

Part 3: The "AI" Twist

The paper is unique because the authors used Artificial Intelligence (specifically ChatGPT) to help solve the hardest part of the puzzle.

The Human-AI Team: The human mathematicians set up the rules of the game (defining the rectangles and triangles). They asked the AI: "Show me a way to arrange dots so we get way more rectangles than triangles."
The Result: The AI spent over an hour thinking and came up with a specific, complex pattern of dots. The humans then checked the math, cleaned it up, and proved it was correct. This pattern was the key to breaking the wave "safety rules."

Part 4: Why This Matters (For Math, Not Medicine)

The paper doesn't talk about curing diseases or building new computers. Its impact is purely in theoretical mathematics and physics:

Breaking a Conjecture: It provides a clear, explicit proof that a famous guess (conjecture) about how waves behave is wrong.
New Counterexamples: Before this, we knew the rules were wrong, but the examples were very abstract and hard to understand. This paper gives a "concrete" example using simple shapes (rectangles and triangles) to show why the rules fail.
Connecting Worlds: It beautifully connects two seemingly unrelated fields: Geometry (counting shapes on a grid) and Analysis (studying how waves move). It shows that the way points are arranged in space dictates the behavior of waves in time.

Summary

In short, the authors played a game of "count the shapes" on a grid. They found a pattern where rectangles vastly outnumber triangles. They then used this pattern to prove that a long-standing theory about how quantum waves behave is incorrect. It's a story about how a simple geometric trick can topple a complex physics rule, solved with a little help from a robot brain.

Technical Summary: Rectangles, Triangles and Schrödinger Waves

Problem Statement
The paper addresses two interconnected problems: a combinatorial geometry question regarding finite subsets of the integer lattice $\mathbb{Z}^2$ , and the validity of weighted $L^2$ estimates for solutions to the Schrödinger equation.

Combinatorial Geometry: The authors investigate whether the number of ordered rectangles in a finite set $A \subseteq \mathbb{Z}^2$ is necessarily controlled by the number of ordered isosceles triangles in $A$ . Specifically, they ask if there exists a constant $M$ such that $\#\square(A) \le M \#\triangle(A)$ (or a "pinned" variant involving the maximum number of triangles sharing a vertex). While upper bounds for these counts exist (e.g., $\#\square(A) \lesssim (\#A)^2 \log \#A$ ), the relationship between the two specific configurations had not been established.
Harmonic Analysis: The paper examines the validity of weighted Strichartz estimates (specifically Stein-type and Mizohata–Takeuchi type estimates) for the Schrödinger equation on both Euclidean space $\mathbb{R}^{d+1}$ and the torus $\mathbb{T}^{d+1}$ . These estimates relate the weighted $L^2$ norm of a solution to the supremum of a weighted X-ray transform of the initial data. Recent work by Cairo established that these estimates fail for general curved hypersurfaces, but the specific failure mechanism and quantitative lower bounds for the paraboloid case remained to be fully characterized, particularly in periodic settings.

Methodology
The authors employ a multi-stage approach combining combinatorial constructions, analytic number theory, and harmonic analysis:

Finite-Field and Lattice Constructions: To address the combinatorial problem, the authors construct finite subsets of $\mathbb{Z}^2$ $Z^{2}$ with a high ratio of rectangles to isosceles triangles. The construction relies on lifting "unit circles" from finite fields $\mathbb{F}_p$ $F_{p}$ (where $p \equiv 3 \pmod 4$ $p \equiv 3 (mod 4)$ ) to the integer lattice via the Chinese Remainder Theorem. The set $A_N(q)$ $A_{N} (q)$ is defined as lattice points in a box $[-N, N]^2$ $[- N, N]^{2}$ that reduce to the unit circle modulo a product of primes $q$ $q$ .
- Note on AI Usage: The authors explicitly state that the specific combinatorial construction achieving the desired asymptotic bounds was discovered using OpenAI's ChatGPT 5.5 Pro, prompted with the geometric definitions and the goal of maximizing the rectangle-to-triangle ratio. The authors then rigorously proved the properties of this construction.
Lattice Lifting and Asymptotics: Using elementary analytic number theory (specifically counting primitive vectors and applying Möbius inversion), the authors lift the counts of geometric configurations from the finite group $\mathbb{Z}_q^2$ to the lattice $\mathbb{Z}^2$ . They establish that for sufficiently large $N$ , the ratio of rectangles to triangles grows polylogarithmically with $N$ .
Transference Principles: The paper establishes a quantitative transference principle between Euclidean and periodic settings. By using a "wave-packet thickening" argument (approximating periodic solutions with band-limited Euclidean solutions), they show that lower bounds for the best constants in periodic estimates imply corresponding lower bounds in the Euclidean setting.
Trigonometric Weights: To improve the lower bounds beyond those derived from simple geometric configuration counts (which yield weights of the form $|u|^2$ ), the authors construct specific non-negative trigonometric polynomial weights. These weights are built from tensor products of one-dimensional subset sums of parabola points, designed to exploit the discrete frequency structure of the torus.

Key Results

Combinatorial Lower Bounds (Theorem 1.2): The authors prove that the ratio of rectangles to isosceles triangles in finite subsets of $\mathbb{Z}^2$ can be arbitrarily large. Specifically, for large $N$ and $n$ , the ratios $R(N)$ and $R_{pin}(N)$ satisfy:
$R(N), R_{pin}(N) \gtrsim \frac{(\log N)^c}{\log \log \log N}$
This provides a negative answer to the question of whether rectangles are controlled by isosceles triangles.
Failure of Periodic and Euclidean Estimates (Theorems 1.3, 1.4, 1.5):
- The combinatorial lower bounds translate directly into lower bounds for the best constants in the periodic Stein-type and Mizohata–Takeuchi estimates in dimension $d=2$ .
- By utilizing the constructed trigonometric weights, the authors establish stronger lower bounds for general dimension $d \ge 1$ . They show that the best constants $ST_d(N)$ and $MT_d(N)$ grow at least as $(\log N)^d$ .
- For the case where the frequency support size $n$ is fixed, the constants grow linearly with $n$ : $\tilde{ST}_d(n), \tilde{MT}_d(n) \sim_d n$ .
- Through the transference principle, these results imply that the Euclidean constants $SR_d(N)$ and $MR_d(N)$ also satisfy the polylogarithmic lower bound $(\log N)^d$ .
Equivalence of Formulations: The paper demonstrates that the periodic and Euclidean formulations of these weighted estimates are mutually equivalent in terms of their failure, although the transference of quantitative lower bounds is primarily from periodic to Euclidean in their construction.

Significance and Claims
The paper claims to provide new explicit combinatorial counterexamples to the Mizohata–Takeuchi conjecture for the paraboloid case. While previous counterexamples by Cairo and others relied on curved hypersurfaces or power-law losses, this work:

Identifies the periodic analogues of the Stein and Mizohata–Takeuchi conjectures.
Provides a transference principle linking periodic arithmetic obstructions to Euclidean analytic estimates.
Reduces the problem of estimating the failure of these analytic inequalities to solvable geometric problems (counting rectangles vs. triangles).
Offers a purely combinatorial explanation for the failure of these estimates, distinct from the geometric curvature arguments used in previous counterexamples.

The authors note that while their lower bounds are weaker than the power-loss bounds found in some recent literature for general hypersurfaces, they are significant because they apply specifically to the paraboloid and are derived from elementary combinatorial structures. The work completes a research program initiated by the first author's EPSRC grant, connecting lattice combinatorics to the analysis of dispersive PDEs.