🔢 mathematics

Bounds on the exceptional set in the $abc$ conjecture

This paper establishes a power-saving bound on the size of the exceptional set of coprime triples $(a,b,c)$ satisfying $a+b=c$ that violate the $abc$ conjecture, utilizing a combination of geometric number theory and Fourier analysis to estimate the density of integer points on high-dimensional varieties.

Original authors: Christian Bernert, Tim Browning, Jared Duker Lichtman, Joni Teräväinen

Published 2026-05-12

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Christian Bernert, Tim Browning, Jared Duker Lichtman, Joni Teräväinen

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 a detective trying to solve a very specific, ancient riddle involving three numbers: a, b, and c.

The riddle is simple: a + b = c.
But there's a catch. These numbers must be "coprime," meaning they don't share any common building blocks (factors) other than 1. Think of them as three unique Lego towers built from completely different sets of bricks.

The Big Mystery: The "abc" Conjecture

Mathematicians have a famous theory called the abc Conjecture. It suggests that if you add two numbers together to get a third, the "building blocks" (prime factors) of the result are usually very "rich" and complex.

To measure this complexity, mathematicians use something called the radical. Imagine the radical as the "unique ingredient list" of a number.

If you have the number 12, its ingredients are 2 and 3 (since $12 = 2 \times 2 \times 3$ ). The radical is $2 \times 3 = 6$ .
If you have 30, the ingredients are 2, 3, and 5. The radical is 30.

The abc Conjecture says: "Usually, the unique ingredient list of the sum ( $c$ ) and its parts ( $a$ and $b$ ) is huge—almost as big as the number $c$ itself."

However, there are exceptions. Sometimes, you find a trio where the ingredient list is surprisingly small compared to the size of the numbers. These are the "bad apples" or the exceptional set. The conjecture claims these bad apples are rare.

The Problem: How Rare?

For a long time, mathematicians knew these bad apples were rare, but they didn't have a good way to count exactly how rare they were as the numbers got bigger.

Think of it like trying to count how many people in a city of 1 million are wearing a specific, weird hat.

The Old Estimate (The "Trivial Bound"): Previous math said, "Well, if the city has 1 million people, maybe up to 100,000 could be wearing that hat." That's a lot of hats! It's a very loose estimate.
The New Discovery: This paper by Bernert, Browning, Lichtman, and Teräväinen says, "Wait, we can do better. We can prove that the number of people wearing that hat is actually much, much smaller than 100,000. It's more like a few hundred."

How They Did It: The "Anatomic" Dissection

The authors didn't just guess; they used a clever strategy to break the problem down, which they call an "anatomic reduction."

Imagine you have a giant, complex machine (the equation $a+b=c$ ). Instead of trying to study the whole machine at once, they took it apart into tiny, individual gears and springs.

Breaking it down: They realized that any number can be broken into a "core" part and several "layered" parts.
The Recipe: They turned the problem of counting these bad number triples into a problem of counting solutions to a very specific, high-dimensional recipe.

The Two Tools in Their Toolbox

To count these solutions, they used two powerful tools from different branches of math:

Fourier Analysis (The "Sound Wave" Tool):
Imagine trying to find a specific pattern in a noisy crowd. Fourier analysis is like using a special microphone that filters out the noise and isolates the specific rhythm you are looking for. The authors used this to show that the "bad" number combinations don't happen as often as they might seem because their patterns cancel each other out.
Geometry of Numbers (The "Grid" Tool):
Imagine plotting all possible number combinations on a giant, multi-dimensional grid. The "Geometry of Numbers" helps you see how many points (solutions) can fit inside a specific shape on that grid without overlapping. They used this to prove that the "bad" triples are squeezed into a very small, tight corner of the grid, meaning there simply isn't enough room for many of them.

The Result: A Tighter Net

By combining these tools, the authors built a much tighter net to catch the "bad apples."

For the general case: They proved that the number of exceptions grows much slower than anyone previously thought. If the old math said the count was proportional to $X^{0.66}$ (roughly), they proved it's actually closer to $X^{0.65}$ or even lower.
For the most interesting case (where the numbers are small): They managed to prove an even stronger result, showing the count is bounded by $X^{0.6}$ .

Why Does This Matter?

In the world of math, getting a "power-saving" improvement (like going from 0.66 to 0.60) is a huge deal. It's like upgrading a security system from catching 90% of thieves to catching 99%.

This paper doesn't solve the whole abc Conjecture (which would require proving there are zero exceptions for certain conditions), but it significantly narrows the search area. It tells us that the "exceptions" are not just rare; they are extremely rare, and we now have a much better map of where to look (or rather, where not to look).

In short: The authors took a messy, hard-to-count problem, broke it into tiny pieces, used sound-wave math and grid-geometry math to filter out the noise, and proved that the "bad" number combinations are far fewer than we ever dared to hope.

Technical Summary: Bounds on the Exceptional Set in the abc Conjecture

Problem Statement
The paper investigates the distribution of solutions to the equation $a + b = c$ , where $a, b, c$ are coprime natural numbers. Specifically, it focuses on counting the number $N_\lambda(X)$ of such "abc triples" with height at most $X$ (i.e., $a, b, c \le X$ ) that satisfy the condition $\text{rad}(abc) < c^\lambda$ , where $\text{rad}(n)$ denotes the product of distinct prime factors of $n$ .

The well-known abc conjecture of Masser and Oesterlé asserts that for any $\lambda < 1$ , the set of such triples is finite. While the best unconditional result (Stewart and Yu) establishes finiteness for $\text{rad}(abc) < (\log c)^{3-\varepsilon}$ , this paper aims to provide quantitative bounds on the size of the exceptional set $N_\lambda(X)$ as $X \to \infty$ . Prior to this work, the only general estimate for $\lambda < 1$ was a "trivial bound" derived from de Bruijn's result on the density of integers with small radicals, yielding $N_\lambda(X) \ll_\varepsilon X^{2\lambda/3 + \varepsilon}$ . The authors seek to improve this exponent, particularly for $\lambda \in (0, 2]$ .

Methodology
The authors employ a combination of analytic number theory tools, specifically Fourier analysis and the geometry of numbers, to bound the number of solutions. The methodology proceeds through the following stages:

Anatomic Reduction: The authors first decompose the variables $a, b, c$ using a lemma (Lemma 2.2) that factorizes any integer $n$ into a small "core" $n_0$ and a product of powers of pairwise coprime integers $n_j^j$ . This reduces the problem of bounding $N_\lambda(X)$ to bounding the number of solutions to monomial Diophantine equations of the form:
$a_0 \prod_{i=1}^d a_i^i + b_0 \prod_{i=1}^d b_i^i = c_0 \prod_{i=1}^d c_i^i$
subject to specific size constraints on the variables. This reduction transforms the original counting problem into estimating the quantity $B_d(a_0, b_0, c_0, A, B, C)$ , which counts solutions within dyadic ranges.
Fourier Analysis: To bound $B_d$ , the authors utilize exponential sums and orthogonality of characters. By applying Hölder's inequality and estimating moments of exponential sums (specifically the second and fourth moments), they derive bounds that depend on the distribution of the variables. This approach recovers the trivial bound but provides the necessary structure to combine with other methods.
Geometry of Numbers: The authors treat the reduced equation as a linear equation by fixing a subset of variables and applying a classical bound for primitive solutions to linear forms (citing Heath-Brown). This yields a bound inversely proportional to the product of the fixed variables, offering a different dependency on the parameters than the Fourier method.
Optimization via Linear Programming: For the case $\lambda < 1$ , the authors combine the bounds derived from the Fourier analysis and the geometry of numbers. They formulate a mixed-integer linear programming problem (solved using Python code provided in the Appendix) to optimize the exponents. This optimization determines the best possible trade-off between the different bounding techniques across various subsets of the decomposed variables.

Key Contributions and Results

Main Theorem (Theorem 1.2): For $\lambda \in (0, 2]$ , the authors establish a power-saving improvement over the trivial bound. They prove:
$N_\lambda(X) \ll_\varepsilon X^{\frac{23\lambda + 3}{40} + \varepsilon}$
This result is non-trivial (i.e., improves upon $X^{2\lambda/3}$ ) for $\lambda \in (9/11, 2]$ . Specifically, for $\lambda \in (0, 1)$ , this yields an exponent of approximately $0.65$.
Refined Result for $\lambda < 1$ (Theorem 1.3): By utilizing the more complex optimization derived from combining the bounds, the authors improve the exponent for the range $\lambda \in (0, 1)$ . They show:
$N_\lambda(X) \ll_{\varepsilon, \lambda} X^{0.6 + \varepsilon}$
This improvement is non-trivial for $\lambda \in (9/10, 1)$ .
Methodological Synthesis: The paper merges techniques from an earlier work by the last three authors (which used the determinant method and Thue equations) with a new approach by the first author. Crucially, the current bounds are achieved without relying on the determinant method, suggesting that the geometry of numbers and Fourier analysis alone can yield significant improvements.

Significance and Claims
The authors state that their primary goal is to provide the first power-saving improvement over the simple bound $X^{2\lambda/3}$ for values of $\lambda \in (0, 2)$ . The significance of the work lies in:

Quantitative Progress: Providing explicit, improved upper bounds on the number of exceptional triples, moving beyond the qualitative finiteness results of the abc conjecture.
Technique Demonstration: Demonstrating that a combination of the geometry of numbers and Fourier analysis, optimized via linear programming, can effectively bound the exceptional set without needing the more complex determinant method or Thue equation theory used in previous refinements.
Modesty on Future Improvements: The authors note that while their bounds do not use the determinant method (implying potential for further improvement), they expect that "a serious numerical improvement will be likely to require new ideas." They acknowledge that beyond their work, no general estimates for $N_\lambda(X)$ with $\lambda < 1$ were previously known beyond the trivial bound.

The paper concludes by providing the Python code used to solve the linear program, ensuring the reproducibility of the optimization step that yields the $0.6$ exponent.