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Quantitative Khintchine on the parabola with non-monotonic approximation functions

This paper establishes a quantitative convergence case of Khintchine's theorem for points on the parabola with non-monotonic approximation functions by deriving explicit constants in classical number theoretic results, particularly Burgess' bound for character sums.

Original authors: Maiken Gravgaard, Simon Kristensen

Published 2026-08-06

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Maiken Gravgaard, Simon Kristensen

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

Imagine you are trying to hit a moving target with a dart, but the rules of the game are incredibly strict. You can only throw darts that land on specific, invisible grid lines drawn across a vast field. The closer your dart lands to the center of a grid square, the better your score. In the world of mathematics, this game is called "Diophantine approximation." It's all about how well we can approximate messy, irrational numbers (like the square root of 2 or pi) using simple fractions (like 22/7). For over a century, mathematicians have been trying to figure out the rules of this game: How close can you get? How many times can you get close? And does it matter if your target is just a random point on a flat sheet of paper, or if it's stuck on a specific shape, like a curved line?

The big question this paper tackles is about a specific shape: a parabola. Think of a parabola as the perfect, smooth curve of a rainbow or the path a ball takes when you throw it. In this mathematical game, the "target" isn't just any point; it's a point that must lie exactly on this curved line. For a long time, mathematicians knew that if you tried to hit this curve with your darts, there were strict limits on how often you could get a "bullseye" (a very close approximation). A famous rule from 1924, known as Khintchine's Theorem, acts like a referee. It says that if your target is hard enough to hit (meaning the "closeness" requirement gets stricter very fast), you will almost never hit it. However, there was a catch: to make this referee's rule work, mathematicians had to assume that your "closeness" requirement got stricter in a very predictable, smooth way—like a ramp that only goes down, never up. This is called being "monotonic."

This paper, written by Maiken Gravgaard and Simon Kristensen, asks a bold question: What if the rules of the game change unpredictably? What if the "closeness" requirement jumps up and down like a rollercoaster instead of sliding down a smooth ramp? For a long time, no one knew if the referee's rule still held true in this chaotic scenario, especially for points stuck on a curved line like a parabola. The authors set out to prove that even if the rules are messy and non-monotonic, the game still has a limit: you still almost never hit the target if the rules get strict enough. They didn't just prove it exists; they calculated the exact, albeit incredibly tiny, numbers that define how close you can get before the game becomes impossible.

The Rollercoaster of Numbers

So, what exactly did these authors do? They took the famous Khintchine theorem and stripped away the "smooth ramp" rule. They wanted to see if the theorem holds up when the approximation function (the rule that tells you how close you need to be) is allowed to be wild and non-monotonic. In the world of math, removing this "smoothness" requirement is like trying to navigate a maze where the walls suddenly shift position. It makes the problem significantly harder.

The authors focused on the parabola, the set of points $(x, x^2)$ . They wanted to know: If we have a list of rules for how close a fraction needs to be to a point on this curve, and those rules jump around wildly, how many points on the curve can actually satisfy them? Their main finding is a resounding "almost none." They proved that if the sum of the squares of these wild rules converges (a fancy way of saying the rules get strict fast enough), then the number of points on the parabola that satisfy them is effectively zero.

But here is the twist: they didn't just say "it's zero." They wanted to be a quantitative referee. They wanted to give a specific number, let's call it $\kappa$ , that represents how strict the rules must be to guarantee that almost no points are hit. They found that such a number exists, but it is so incredibly small that it feels almost comical.

The Monster Constants

To get these numbers, the authors had to wrestle with some very old, very stubborn mathematical tools. They used a technique involving "character sums," which are like adding up waves of numbers to see if they cancel each other out. To estimate these sums, they relied on a famous bound discovered by mathematician Burgess. However, the standard version of Burgess's bound wasn't precise enough for their needs. They needed an "explicit" version, meaning they needed to know the exact size of the constants involved, not just that they existed.

This is where the paper gets a bit wild. The authors had to calculate these constants for different types of numbers (primes, composite numbers, large numbers, small numbers). The result is a set of four different theorems, each with its own version of the constant $\kappa$ .

In their most general version (Theorem 4), which works for any denominator $q$ , the constant $\kappa$ is a nightmare of tiny numbers. One of the terms in their calculation is roughly $10^{-1023.89775276641198}$ . To put that in perspective, if you wrote that number out, it would have over a thousand zeros after the decimal point before you even got to the first non-zero digit. It is so small that it is practically zero, yet mathematically, it is the key that unlocks the proof. The authors admit that this number is "very small" and that the main culprit is the "divisor function," which counts how many ways a number can be divided. Because this function can get huge for certain numbers, it forces their constant to shrink to almost nothing.

However, the authors didn't just leave it at that. They realized that if they made the game slightly more specific, they could get much more "sane" numbers.

The "Large $q$ " Version (Theorem 5): If they only look at very large denominators (specifically, $q$ larger than $e^{e^{41}}$ , a number so big it's hard to comprehend), the constant $\kappa$ jumps up to a much more reasonable size, around $0.00499$.
The "Prime" Version (Theorem 7): If they only look at denominators that are prime numbers, the constant improves again, reaching about $0.012$.
The "Few Divisors" Version (Theorem 6): If they look at numbers that don't have too many factors, the constant sits somewhere in between.

Why This Matters (Even if the Numbers are Weird)

You might wonder, "Who cares about a number that is $10^{-1023}$ ?" The answer lies in the structure of the proof. Before this paper, we didn't know if a constant existed at all for non-monotonic functions on a parabola. The fact that the authors could prove it exists, even if the number is microscopic, is a massive step forward. It confirms that the "curved" nature of the parabola doesn't magically allow us to bypass the rules of approximation, even when the rules are chaotic.

The paper also highlights a specific problem in the mathematical toolkit. The authors point out that the reason their numbers are so tiny is due to the "divisor function" bounds they had to use. They suggest that if mathematicians can find better ways to estimate how many divisors a number has, these constants could become much larger and more useful. They essentially built a bridge across a canyon, but the bridge is made of a material that is so thin it's almost invisible. It proves the bridge can exist, but it also tells us we need to find a stronger material to make it walkable.

In the end, Gravgaard and Kristensen have shown that the parabola is a stubborn opponent. Whether you approach it with smooth, predictable rules or chaotic, jumping rules, it resists being approximated. They have provided the mathematical proof that this resistance is absolute, quantifying the limits of our ability to hit the target with a precision that is both terrifyingly small and rigorously exact. They haven't solved the problem of making the numbers bigger, but they have definitively shown that the game is unwinnable under these conditions, no matter how wild the rules get.

Technical Summary: Quantitative Khintchine on the Parabola with Non-Monotonic Approximation Functions

Problem Statement
The paper addresses a quantitative version of the convergence case of Khintchine's theorem in metric Diophantine approximation, specifically restricted to points lying on the parabola $P := \{(x, x^2) \mid x \in [0, 1]\}$ . The central challenge is to remove the standard assumption that the approximation function $\psi: \mathbb{N} \to [0, \infty)$ must be monotonic.

While Khintchine's theorem (1924) establishes a zero-one law for the Lebesgue measure of $\psi$ -approximable points in $\mathbb{R}^n$ , the divergence case strictly requires monotonicity. The convergence case, however, does not inherently require it for the ambient space, yet extending this to manifolds (like the parabola) without monotonicity has proven difficult. Previous results for non-degenerate submanifolds, such as those by Beresnevich and Yang, rely on monotonicity even for convergence. A notable exception is Huang's theorem (2023), which proved the convergence case for the parabola without monotonicity but did not provide explicit quantitative bounds on the measure of the set of well-approximable points.

The authors aim to provide explicit constants for the measure of the set of points $x \in [0, 1]$ such that the simultaneous approximation error $\max(\|qx\|, \|qx^2\|)$ exceeds a fixed multiple $\kappa \psi(q)$ for all $q$ . Specifically, they seek to establish that for a convergent series $S_\psi = \sum \psi(q)^2$ , the measure of the set $B'(\psi, \kappa) = \{x \in [0, 1] : \max(\|qx\|, \|qx^2\|) > \kappa \psi(q) \forall q \in \mathbb{N}\}$ is at least $1 - \delta$ , with explicit dependence of $\kappa$ on $\delta$ and $S_\psi$ .

Methodology
The proof strategy follows the framework established by Huang [13] but focuses heavily on deriving explicit constants throughout the estimation process. The methodology proceeds in the following stages:

Reduction to Counting: The problem is reduced to counting the number of rational points $a/q$ with a fixed denominator $q$ that lie close to the parabola. This is formulated as estimating the counting function $\sum_{a \le q, \|a^2/q\| \le 3\kappa\psi(q)} 1$ .
Exponential Sums and Fejér Kernels: The counting function is estimated using a Fejér kernel to smooth the indicator function of the interval. This transforms the problem into bounding exponential sums of the form $\sum_{a=1}^q e^{2\pi i j a^2 / q}$ .
Dirichlet Characters and Gauss Sums: These quadratic exponential sums are rewritten in terms of Dirichlet characters (specifically Legendre and Jacobi symbols) using properties of quadratic Gauss sums.
Explicit Burgess Bounds: The core technical innovation lies in the application of Burgess's bound for short character sums. Since standard effective versions of Burgess's bound do not hold for all composite moduli, the authors derive explicit bounds for three distinct ranges of the modulus $q$ $q$ :
- Small $q$ ( $1 \le q \le 10^5$ ): Constants are calculated via brute-force computation.
- Medium $q$ ( $10^5 < q \le \exp(\exp(8))$ ): Bounds are derived using results from Bordignon [7], with explicit verification of the conditions.
- Large $q$ ( $q > \exp(\exp(8))$ ): Bounds rely on results from Bordignon [7] and Robin [21].
Divisor Function Estimates: A significant portion of the work involves managing the divisor function $d(q)$ , which appears in the error terms. The authors utilize explicit bounds for $d(q)$ (e.g., $d(q) \le K_\beta q^\beta$ ) and analyze how these bounds affect the final constants. They address the issue of large divisor values by considering specific subsets of integers (e.g., numbers with few prime factors or very large numbers) to obtain more reasonable constants.

Key Contributions and Results
The paper presents four main theorems providing quantitative bounds for $\kappa(\delta)$ under different constraints on the denominator $q$ . In all cases, if $S_\psi = \sum \psi(q)^2$ converges, then $\lambda(B'(\psi, \kappa)) \ge 1 - \delta$ .

Theorem 4 (All $q$ ): Provides a bound for all integers $q$ . The resulting constant $\kappa(\delta)$ is extremely small (on the order of $10^{-1023}$ for certain terms) due to the worst-case behavior of the divisor function $d(q)$ in the general case.
Theorem 5 (Large $q$ ): Assumes $\psi(q) = 0$ for $q \le \exp(\exp(41))$ . By restricting to very large moduli, the authors utilize improved bounds for $d(q)$ (specifically $d(q) \le q^{1/57}$ ), resulting in significantly more manageable constants (e.g., $\kappa \approx 0.00499 (\delta/S_\psi)^{2/3}$ ).
Theorem 6 (Bounded Prime Factors): Assumes $q$ has at most $k$ prime divisors. This restriction limits $d(q) \le 2^k$ , yielding constants that depend on $k$ but are generally more reasonable than Theorem 4.
Theorem 7 (Prime Moduli): Assumes $q$ is prime. This case utilizes the best explicit Burgess bounds available for prime moduli, yielding the most favorable constants among the general cases (e.g., $\kappa \approx 0.012 (\delta/S_\psi)^{2/3}$ ).

Significance and Claims
The authors claim that this work is the first to provide a quantitative Khintchine theorem for the parabola with non-monotonic approximation functions. The primary significance lies in the explicit nature of the constants, which allows for the formulation of problems regarding the dependency between the approximation constant $\kappa$ and the measure of the set of points, a problem previously raised by Adiceam et al. [1].

The paper acknowledges that the constants, particularly in the most general case (Theorem 4), are very small. The authors attribute this limitation not to the method of proof itself, but to the explicit constants arising from the divisor function bounds within the variants of Burgess's theorem used. They note that while better estimates are desirable, achieving them with the current methods is unlikely. The work serves as a foundational step, demonstrating that non-monotonic convergence results can be made quantitative, albeit with the caveat of small constants in the general setting.

The Rollercoaster of Numbers

The Monster Constants

Why This Matters (Even if the Numbers are Weird)

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