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Area correlations related to lattice points in discs

Motivated by the Lester-Wigman conjecture, this paper demonstrates that circle flows on tori exhibit "mixing for the area observable" and determines the probability density functions of areas in both global and local regimes, despite the flow being essentially integrable.

Original authors: Matteo Bordignon, Pär Kurlberg

Published 2026-07-16

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Matteo Bordignon, Pär Kurlberg

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 Great Grid Game: When Circles Meet Squares

Imagine you are looking at a giant, infinite floor tiled perfectly with square tiles, like a checkerboard that never ends. Now, picture drawing a massive circle on top of this floor. As the circle gets bigger and bigger, it slices through the edges of these tiles. Some tiles are almost entirely inside the circle, some are almost entirely outside, but the ones the circle cuts through are the interesting ones. These "boundary tiles" get chopped into weird shapes, and the amount of tile that ends up inside the circle is a fraction between 0 and 1.

This scenario sits at the intersection of geometry and number theory, two fields that often feel like they speak different languages. One asks, "What does this shape look like?" while the other asks, "How many whole numbers fit inside it?" For over a century, mathematicians have been fascinated by the "Gauss Circle Problem," which is essentially a game of counting how many grid points (the corners of the tiles) fall inside a circle. But there's a twist: the error in that count isn't random noise; it has a hidden structure. Recently, researchers started asking a deeper question: If you look at the slices of the tiles along the edge of the circle, do the sizes of those slices behave like a chaotic, unpredictable mess, or is there a hidden order? Specifically, if you know the size of the slice in one tile, can you guess the size of the slice in a tile that is a significant distance away along the curve? This paper dives into that question, treating the circle's edge not just as a shape, but as a moving path that sweeps across the grid.

The Story of the Sweeping Circle

In this paper, Matteo Bordignon and Pär Kurlberg take us on a journey to see if the "slices" of these grid tiles are connected or if they are essentially independent of each other. They investigate a specific idea called the "vanishing area correlation conjecture." In plain English, this conjecture suggests that as the circle grows infinitely large, the size of the slice in one tile has absolutely no influence on the size of the slice in another tile, even if you look at tiles that are a macroscopic distance apart. Specifically, the paper proves this holds true as long as the distance between the two tiles (measured in the number of tiles along the circle's edge) is roughly proportional to the circle's radius raised to a power slightly less than 1. This is a vast distance, covering a significant fraction of the circle's total circumference, rather than just immediate neighbors. It's like flipping a coin: if you flip heads, it doesn't change the odds of the next flip being heads or tails, even if you wait for a long sequence of flips to pass.

The authors prove that this "no connection" idea is true. They show that the correlation between the areas of these slices does indeed vanish (disappear) as the circle gets huge. The authors describe this as "mixing for the area observable," a fancy way of saying the system forgets its past quickly in terms of area statistics, even though the circle's path is actually very predictable and orderly. In fact, the paper explicitly states that the flow is "essentially integrable" and that true "mixing" (in the strict dynamical systems sense) certainly does not hold because the angle of the circle's path relative to the grid lines remains essentially constant. However, despite this rigid order, the specific area measurements behave as if they are random and uncorrelated over these large distances.

To understand how they did this, imagine the circle's edge as a laser beam sweeping across the floor. The authors realized that while the beam is moving, it doesn't just hit tiles randomly. It follows a smooth curve. However, they found a clever trick: if you look at the beam over a very short distance, the curve is so smooth that it looks like a straight line. They used a "probabilistic model" to pretend the circle is actually a family of straight lines with slightly different angles. By doing this, they could show that as the angle of the line changes just a tiny bit, the amount of tile it cuts through sweeps through every possible size from zero to one, completely randomly. Because the angle changes so smoothly and covers all possibilities, the specific size of the cut in one spot becomes completely unrelated to the size of the cut a significant distance further down the line.

The paper also calculates exactly what the distribution of these slice sizes looks like. They found that the sizes aren't spread out evenly. Instead, there are "singularities," or spikes, at the very ends. This means that it is actually quite common for the circle to slice off a tiny sliver of a tile (almost zero area) or a huge chunk (almost the whole tile), and less common for it to cut the tile right down the middle. They even provided a precise mathematical formula for the probability of getting a slice of any specific size, which involves some complex integrals but essentially describes a curve that dips in the middle and spikes at the edges.

One of the most interesting parts of their work is how they handled the "bad" spots. In the world of circles and grids, there are certain angles where the circle's edge runs almost perfectly parallel to the grid lines (specifically, where the slope is well-approximated by simple fractions). In these specific spots, the slices might not behave randomly. The authors didn't simply exclude these spots from their study; instead, they proved that these tricky spots are so rare that they are negligible in number compared to the total number of tiles. They showed that the result holds for "most" intervals, and the contribution from these rare, non-random angles is so small that it doesn't ruin the overall picture. As long as you look at a large enough section of the circle, the "bad" spots are just a tiny drop in the ocean, and the rest of the water flows perfectly smoothly.

The authors also looked at the average size of these slices. They confirmed that, on average, the circle cuts exactly half of each boundary tile. This might sound obvious, but proving it rigorously for a growing circle is a significant achievement. They also calculated the "variance," which measures how much the slice sizes wiggle around that average. They found a specific number, approximately 0.119, that describes this wiggle room for the entire circle.

In the end, this paper confirms that even though a circle moving across a grid is a very orderly, predictable system (mathematicians call this "integrable"), the way it cuts up the tiles looks surprisingly random over large distances. The area of the slice in one tile doesn't predict the area in another tile a macroscopic distance away. It's a beautiful example of how order can create the illusion of chaos, and how mathematicians can use the tools of probability to understand the behavior of rigid geometric shapes. The authors didn't just guess this; they proved it with rigorous math, showing that the "memory" of the circle's path vanishes as it grows, leaving behind a pattern that is as random as a roll of the dice.

Technical Summary: Area Correlations Related to Lattice Points in Discs

Problem Statement
Motivated by the Lester-Wigman vanishing area correlation conjecture regarding lattice points near the boundary of circles with growing radius, this paper investigates the statistical properties of the areas of intersection between a large disc $B(R)$ and unit squares $S_\lambda$ centered at lattice points $\lambda \in \mathbb{Z}^2$ . Specifically, the authors study the correlation of the "area observable" $A(\lambda)$ , defined as the area of $S_\lambda \cap B(R)$ , for lattice points ordered by their argument along the circle's boundary.

The central question is whether the correlation between $A(\lambda_j)$ and $A(\lambda_{j+k})$ vanishes as the radius $R \to \infty$ , even when the separation $k$ grows with $R$ (specifically $k = O(R^{1-\epsilon})$ ). This problem is framed within the context of circle flows on tori, which can be physically interpreted as the motion of a charged particle in a magnetic field, where $R \to \infty$ corresponds to a vanishing magnetic field. While such flows are essentially integrable and not mixing in the traditional sense, the authors investigate whether "mixing for the area observable" still occurs.

Methodology
The authors employ a hybrid approach combining dynamical systems, analytic number theory, and probabilistic modeling:

Reduction to Local Regimes: The global correlation sum is decomposed into local sums over small angular intervals $I$ . The analysis is reduced to the seventh octant of the circle, where the tangent slope lies in $(0, 1)$ , utilizing the symmetries of the square lattice $\mathbb{Z}^2$ .
Probabilistic Model of Circle Intersections: The authors construct a probabilistic model for the family of circles $C(R)$ intersecting the fundamental domain $S = [0, 1)^2$ . They define a measure $\mu$ on the space of circles passing through the "lower/left" boundary $\partial_L(S)$ of the unit square, parameterized by the entry point $v$ and the tangent angle $\theta$ .
Approximation by Rational Slopes: To handle the distribution of lattice points, the circular arc is approximated by tangent lines, which are further approximated by lines with rational slopes derived from Farey fractions. This allows the authors to utilize equidistribution results for sequences of the form $\alpha + \frac{p}{q}n \pmod 1$ .
Order Stability and Decorrelation: A key technical insight is the exploitation of "order stability." The authors show that for a fixed displacement vector $\lambda$ between lattice points, the area of the first intersection is essentially constant, while the area of the second intersection "sweeps through" all possible values uniformly as the tangent angle varies slightly. This decoupling allows the correlation sum to be approximated by an integral where the terms effectively decorrelate.
Exclusion of "Bad" Intervals: The analysis carefully excludes angular intervals where the tangent slope is well-approximated by rational numbers of low height (Diophantine conditions), showing that the contribution of such intervals to the total sum is negligible ( $o(R|I|)$ ).

Key Contributions and Results

Vanishing Area Correlation (Theorem 1): The primary result establishes that the global area correlation vanishes. Specifically, for $k = O(R^{1-\epsilon})$ and angular intervals $I$ satisfying specific size constraints (avoiding intervals where the tangent slope is too well approximated by low-height rationals), the correlation $\text{Corr}(R, k, I)$ tends to zero as $R \to \infty$ . This confirms an analogue of the Lester-Wigman conjecture in this setting, demonstrating "mixing for the area observable" despite the underlying flow being integrable.
Asymptotics of Mean and Variance (Proposition 2):
- The global mean area is shown to be $\text{Mean}(R) = 1/2 + o(1)$ .
- The global variance converges to a positive constant: $\text{Var}(R) \to c_{\text{glob}} \approx 0.119$ , where $c_{\text{glob}} = \frac{13}{60} - \frac{1}{12}\log(1+\sqrt{2}) - \frac{1}{30\sqrt{2}}$ .
- The paper notes that the $1/2 + o(1)$ mean implies a slight improvement on Gauss's $O(R)$ error term for the circle problem due to a "perfect cancellation" of boundary terms.
Local Statistics: The authors determine the local mean and variance. While the local mean remains $1/2 + o(1)$ for intervals of size $|I| > R^{-1/2+\epsilon}$ , the local variance depends on the specific angle $\theta$ of the interval, converging to a function $g(\theta)$ derived from the geometry of line intersections with the square.
Limiting Distribution (Proposition 3): The paper derives the probability density function (PDF) $f(a)$ for the area $A(\lambda)$ as $R \to \infty$ . The density is given by an integral involving elliptic functions and exhibits two integrable singularities at $a=0$ and $a=1$ . The explicit formula is:
$f(a) = \frac{2z_a}{\sqrt{1+4z_a^2}} + \frac{1}{\sqrt{2}z_a} \int_{1}^{2z_a\sqrt{m}} \frac{1}{(1+m^2)^{3/2}} dm$
where $z_a = \min(a, 1-a)$ .

Significance and Claims
The paper claims to resolve the vanishing area correlation conjecture for lattice points on circles in a setting related to circle flows on tori. The significance lies in demonstrating that even in an essentially integrable system (where global mixing does not hold), specific observables (the area of intersection) can exhibit mixing behavior (vanishing correlations).

The authors note that their results provide a probabilistic model for the distribution of areas, which is distinct from the setup in the original Lester-Wigman work (where squares were either fully inside or outside the disc). Furthermore, the paper suggests that if the areas $\{A(\lambda)\}$ were independent and identically distributed (IID), an error term of $O(R^{1/2+\epsilon})$ for the Gauss circle problem would follow, though the authors do not claim to prove the IID hypothesis itself.

The work is presented as a study of the dynamics of circle flows, offering a new perspective on the Gauss circle problem by dividing intersecting squares into internal and boundary categories and estimating boundary counts via the length of the boundary, echoing Gauss's original argument.

The Great Grid Game: When Circles Meet Squares

The Story of the Sweeping Circle

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