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The inhomogeneous Khintchine Theorem in dimension two

This paper proves that the inhomogeneous variant of Khintchine's Theorem holds in dimension two without requiring a monotonicity assumption, thereby resolving the final open case in the metric theory of inhomogeneous Diophantine approximation and aligning it with the homogeneous counterpart.

Original authors: Demi Allen, Manuel Hauke-Treuer, Felipe A. Ramírez

Published 2026-05-20

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Demi Allen, Manuel Hauke-Treuer, Felipe A. Ramírez

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 Catching Falling Rain

Imagine you are standing in a field (this is your "space," specifically a two-dimensional square). Above you, rain is falling. But this isn't normal rain; it's "rational rain." The drops fall at specific, predictable intervals determined by numbers.

The Goal: You want to know if a specific spot on the ground (a point called $\alpha$ ) will get wet infinitely many times.
The Rule: A spot gets wet if a raindrop lands within a tiny distance of it. The size of this "wet zone" is determined by a function called $\psi$ (think of this as the size of the umbrella or the splash radius).
The Twist (Inhomogeneous): In the classic version of this game, the rain falls straight down. In this paper's version, the rain is "shifted" or "tilted" by a fixed, random amount (called $\gamma$ ). It's as if the whole sky is slightly off-center.

The authors are trying to solve a 100-year-old puzzle: If the rain is heavy enough (mathematically speaking), will every spot on the ground get wet infinitely often, even if the rain is tilted?

The Problem: The Missing Piece of the Puzzle

For decades, mathematicians knew the answer to this question in two scenarios:

1D (A line): If the rain is tilted, you must assume the rain gets smaller in a very smooth, predictable way (monotonicity) to guarantee the ground gets wet. If the rain size jumps around randomly, the ground might stay dry.
3D and higher (A room or more): Even if the rain is tilted and the size jumps around randomly, the ground always gets wet. The extra dimensions act like a safety net.

The Mystery: What about 2D (a flat sheet of paper)?
For a long time, this was the "missing link." No one knew if the "safety net" of higher dimensions worked for a flat sheet, or if the 1D rule (needing smooth rain) still applied. This paper finally solves that mystery.

The Discovery: The authors prove that in 2D, the safety net works! You do not need the rain to be smooth or predictable. Even if the rain size jumps around wildly, as long as the total amount of rain is infinite, every spot on the ground will get wet infinitely many times.

The Obstacle: Why Was 2D So Hard?

To prove this, the authors had to count how often two different raindrops might land on the same spot at the same time (overlapping).

The Easy Way (3D+): In higher dimensions, the raindrops are so spread out that they rarely overlap in a messy way. You can use a simple counting trick (like counting how many people are in a room) to prove the ground gets wet.
The Hard Way (2D): In 2D, the raindrops are crowded. They overlap in complex patterns. If you try to use the simple counting trick, the math breaks down because the overlaps are too frequent and chaotic. It's like trying to count how many times two people in a crowded dance floor bump into each other; it's too messy to just guess.

The Solution: "Surgery" and "Shift-Reduction"

The authors developed a clever new strategy to clean up the mess. They call it "Shift-Reduction."

Imagine you are trying to count the overlaps, but the crowd is too chaotic.

The Old Strategy (Rational Shifts): If the tilt ( $\gamma$ ) was a simple fraction (like 1/2), mathematicians could just "cut out" the people who were standing in the wrong spots (non-primitive points). This cleaned up the crowd enough to count the overlaps.
The New Problem (Irrational Shifts): But what if the tilt is a weird, irrational number (like $\pi$ )? You can't just cut out a simple fraction. The crowd remains messy.
The New Strategy (The "Moving Target"): The authors realized they couldn't use one fixed cut. Instead, they had to use a sequence of approximations.
- They pretend the weird tilt is actually a series of simpler fractions that get closer and closer to the real thing.
- For each step in this sequence, they perform a tiny bit of "surgery" on the crowd, removing the people who don't fit that specific approximation.
- They split the problem into different "regimes" (like different weather conditions). Sometimes the approximation is good enough to use the simple cut; other times, they have to use a more delicate, two-step cut.

The Metaphor: Imagine trying to take a photo of a fast-moving, blurry object.

In 1D, you need a very steady hand (monotonicity) to get a clear picture.
In 3D, the object is so big and slow that any camera works.
In 2D, the object is moving fast and is blurry. The authors invented a new camera technique: they take a series of photos, each with a slightly different focus setting (the "shift-reduction"), and then stitch them together. By combining these slightly different views, they can prove the object is there, even though no single photo was perfect.

The Result: A Complete Theory

By using this "shift-reduction" technique, the authors managed to prove that the overlaps in 2D are actually well-behaved enough to guarantee the result.

In simple terms:
They proved that for two-dimensional numbers, the "chaos" of the rain size doesn't matter. The ground gets wet. This completes the story for all dimensions:

1D: You need smooth rain.
2D and up: You don't need smooth rain; the ground always gets wet.

This brings the "inhomogeneous" theory (tilted rain) into perfect alignment with the "homogeneous" theory (straight rain), showing that the dimension of the space is the only thing that matters, not the complexity of the rain's size.

Summary of the "Heuristics" (The Logic)

The Goal: Prove that a set of points has "full measure" (meaning, almost every point in the square is covered).
The Tool: They use a mathematical rule called the Second Borel-Cantelli Lemma. This rule says: "If the sum of probabilities is infinite, and the events aren't too dependent on each other, then the event happens infinitely often."
The Hurdle: In 2D, the events (raindrops landing) are too dependent on each other. They overlap too much.
The Fix: They "surgically" remove the problematic overlaps by restricting the points to a special subset (the "shift-reduced" sets).
The Breakthrough: They showed that even with the messy, irrational tilt, these special subsets are still large enough to cover the ground, and their overlaps are small enough to satisfy the math rule.

Final Takeaway: This paper closes the last open chapter in a century-old mathematical story about how well we can approximate numbers, proving that in two dimensions, nature is more forgiving than we thought.

Technical Summary: The Inhomogeneous Khintchine Theorem in Dimension Two

Problem Statement
The paper addresses the metric theory of inhomogeneous Diophantine approximation, specifically focusing on the simultaneous approximation of two real numbers $\alpha_1, \alpha_2$ by shifted rational numbers $(u_1+\gamma_1)/q$ and $(u_2+\gamma_2)/q$ . The central object of study is the set $W^\gamma(\psi)$ of vectors $\alpha \in [0,1]^2$ such that $\|q\alpha - u - \gamma\| < \psi(q)$ for infinitely many $(u, q) \in \mathbb{Z}^2 \times \mathbb{N}$ , where $\gamma \in \mathbb{R}^2$ is a fixed inhomogeneous parameter and $\psi: \mathbb{N} \to [0, \infty)$ is an approximation function.

The authors aim to prove the "divergence part" of the inhomogeneous Khintchine theorem in dimension $m=2$ without assuming that $\psi$ is monotonic. While the convergence part follows directly from the Borel–Cantelli lemma, the divergence part (establishing that the set has full Lebesgue measure when $\sum \psi(q)^2 = \infty$ ) has historically required monotonicity. This assumption was known to be unnecessary for dimensions $m \ge 3$ (Gallagher, Ennola) and necessary for $m=1$ (Duffin–Schaeffer). The case $m=2$ remained the last open case in the general inhomogeneous theory, as well as the final outstanding case for Khintchine–Groshev-type theorems for systems of linear forms.

Methodology and Obstacles
The proof strategy relies on establishing "quasi-independence on average" for the sets $E_q$ defined by the approximation condition. Standard approaches using the second Borel–Cantelli lemma require bounding the overlap $\lambda(E_q \cap E_r)$ .

The Obstacle: In dimension $m=2$ , elementary overlap estimates involving the greatest common divisor (GCD) of $q$ and $r$ yield a bound that includes a $\log \log Q$ factor. This bound is insufficient to prove quasi-independence for arbitrary $\psi$ , particularly when $\psi$ grows slowly or is sparse. The authors note that previous methods relying on "surgery" (restricting to primitive points) work for rational $\gamma$ or under extra divergence assumptions but fail for general irrational $\gamma$ due to the unbounded growth of the denominators in rational approximations.
Shift Reduction: The authors introduce a refined "shift-reduction" scheme. Instead of a single primitive condition, they define subsets $E'_q \subseteq E_q$ $E_{q}^{'} \subseteq E_{q}$ based on the primitivity of $B_k u + A_k$ $B_{k} u + A_{k}$ (or $b_k u + a_k$ $b_{k} u + a_{k}$ ), where $A_k/B_k$ $A_{k} / B_{k}$ and $a_k/b_k$ $a_{k} / b_{k}$ are rational approximations to $\gamma$ $γ$ derived from the size of $\psi(q)$ $ψ (q)$ .
- If the denominator $B_k$ of the best approximation is small relative to $\psi(q)$ , they use $A_k/B_k$ .
- If $B_k$ is large, they utilize a secondary approximation $a_k/b_k$ provided by Dirichlet's theorem.
Refined Overlap Estimates: By combining this shift-reduction with a "delicate view" of the overlap estimates, the authors derive bounds that include indicator functions depending on the distance of $(q-r)\gamma/\gcd(q,r)$ to the nearest integer. This allows them to exploit the equidistribution properties of $\gamma$ .
Bohr Set Bounds: The core technical innovation involves estimating the cardinality of "Bohr sets" (sets of integers $h$ $h$ such that $\|h\gamma\|$ $∥ hγ ∥$ is small) within specific ranges. The authors handle three distinct regimes based on the Diophantine properties of $\gamma$ $γ$ and the relative sizes of the parameters:
- Rational $\gamma$ : Handled by existing literature (Kim).
- Non-Liouville $\gamma$ : The equidistribution of $h\gamma$ provides sufficient saving.
- Liouville $\gamma$ : This is the most difficult case. The authors employ a case distinction based on the size of the approximation denominator $B$ relative to the support size $M$ . They show that by switching between the primary approximation $A/B$ and the secondary $a/b$ depending on whether $B$ is "small" or "large" relative to $M$ , they can control the overlap estimates uniformly, regardless of how well $\gamma$ is approximated by rationals.

Key Contributions and Results
The primary result is Theorem 1.1, which states that for any $\gamma \in \mathbb{R}^2$ and any function $\psi: \mathbb{N} \to [0, \infty)$ :
$\lambda(W^\gamma(\psi)) = \begin{cases} 0 & \text{if } \sum_{q=1}^\infty \psi(q)^2 < \infty, \\ 1 & \text{if } \sum_{q=1}^\infty \psi(q)^2 = \infty. \end{cases}$
This result holds without any monotonicity assumption on $\psi$ .

The paper further establishes:

Multivariate Inhomogeneous Khintchine–Groshev Theorem: By combining Theorem 1.1 with reduction arguments from previous work (specifically [39]), the authors confirm the theorem for systems of linear forms in all dimensions $(n, m)$ where $m > 1$ , resolving the $(n, 2)$ cases.
Univariate Inhomogeneous Khintchine–Groshev Theorem: The result extends to the univariate case where $\psi$ depends only on $|q|$ .
Hausdorff Measure Dichotomies: Using the Mass Transference Principle, the authors deduce corresponding results for Hausdorff measures, completing the metric theory for these dimensions.

Significance
The paper claims to resolve the "last remaining case" in the metric theory of inhomogeneous Diophantine approximation. By proving the theorem for dimension $m=2$ without monotonicity, the authors bring the inhomogeneous theory into full alignment with its homogeneous counterpart. This confirms a conjecture by the first and third authors regarding the Khintchine–Groshev theory for systems of linear forms. The work demonstrates that the obstruction to removing monotonicity in dimension two is not structural (as it is in dimension one) but technical, and it is overcome through a sophisticated interplay of shift-reduction techniques and precise Bohr set estimates that accommodate both Liouville and non-Liouville parameters.