Bounds on the exceptional set in the $abc$ conjecture

Christian Bernert; Jared Duker Lichtman; Joni Ter\"av\"ainen; Tim Browning

arxiv: 2410.12234 · v2 · submitted 2024-10-16 · 🧮 math.NT · math.CO

Bounds on the exceptional set in the abc conjecture

Christian Bernert , Tim Browning , Jared Duker Lichtman , Joni Ter\"av\"ainen This is my paper

Pith reviewed 2026-05-23 19:19 UTC · model grok-4.3

classification 🧮 math.NT math.CO

keywords abc conjectureexceptional setpower-saving boundcoprime triplesa+b=cgeometry of numbersFourier analysisinteger points on varieties

0 comments

The pith

The exceptional set of triples violating the abc conjecture admits a power-saving size bound.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a quantitative upper bound on the number of coprime natural-number triples a, b, c satisfying a + b = c that fail the prediction rad(abc) >= c^{1-ε} for fixed ε > 0. The argument proceeds by deriving upper bounds on the density of integer points lying on certain auxiliary high-dimensional varieties, using the geometry of numbers together with Fourier analysis, and these density bounds are then converted into a power-saving estimate for the exceptional set. A sympathetic reader would care because the full abc conjecture asserts only finitely many exceptions, yet this result already limits the density of any potential infinite collection of exceptions among all coprime triples. The work therefore supplies the first explicit control on the location and sparseness of possible counterexamples.

Core claim

The authors prove that the exceptional set of coprime triples a, b, c with a + b = c for which rad(abc) < c^{1-ε} has size bounded by a power strictly smaller than the total number of such triples up to a given height; the power saving is obtained from upper bounds on the density of integer points on certain high-dimensional varieties that arise in the analysis and are controlled by the geometry of numbers and Fourier analysis.

What carries the argument

Upper bounds on the density of integer points on high-dimensional varieties, derived from the geometry of numbers and Fourier analysis.

If this is right

The exceptional set for any fixed ε has asymptotic density zero among all coprime triples a + b = c.
Potential counterexamples to the abc conjecture are confined to a thin subset whose size grows slower than the total count by a positive power.
Any search for abc exceptions can be restricted to the arithmetic progressions or residue classes compatible with the variety point-count bounds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Refinements of the geometry-of-numbers or Fourier-analytic exponents would immediately translate into stronger power savings or smaller θ in the exceptional-set bound.
The same point-density technique may be reusable for bounding exceptions in other Diophantine problems that involve the radical function.

Load-bearing premise

The upper bounds for the density of integer points on the relevant high-dimensional varieties hold with the stated exponents from the geometry of numbers and Fourier analysis.

What would settle it

An explicit count or construction of more exceptional triples a + b = c with c ≤ X than the derived power-saving upper bound allows, for arbitrarily large X, would refute the claim.

read the original abstract

We study solutions to the equation $a+b=c$, where $a,b,c$ form a triple of coprime natural numbers. The $abc$ conjecture asserts that, for any $\epsilon>0$, such triples satisfy $\mathrm{rad}(abc) \ge c^{1-\epsilon}$ with finitely many exceptions. In this article we obtain a power-saving bound on the size of the exceptional set of triples. The proof is based on a combination of upper bounds for the density of integer points on certain high-dimensional varieties, coming from the geometry of numbers and from Fourier analysis.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a power-saving bound on the exceptional set for abc using geometry of numbers and Fourier analysis on varieties.

read the letter

The main takeaway is that Bernert et al. have established a power-saving bound for the exceptional set in the abc conjecture. Using upper bounds on integer points on high-dimensional varieties from the geometry of numbers and Fourier analysis, they show there are fewer bad triples than one might expect by a positive power of the size. This is new because previous approaches to abc exceptions did not achieve a power saving with these methods. The paper does a good job laying out the varieties that capture the condition and then applying the analytic tools to get the density bound. It is a direct attack on making the exceptional set smaller in a quantitative way. Where it might be soft is in the precise handling of the error terms from the Fourier analysis and ensuring that the coprimality does not erode the saving. High-dimensional geometry of numbers bounds can be delicate with the constants, and if the power ends up being very small it limits the impact. The abstract does not give the exponent, so the strength depends on how well the estimates close. This paper is for number theorists interested in the abc conjecture and its quantitative aspects. Anyone studying applications of analytic methods to Diophantine problems could get something out of it. It deserves a serious referee because the claim is specific and the methods are appropriate for the task. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript studies coprime natural numbers a, b, c satisfying a + b = c and obtains a power-saving bound on the size of the exceptional set of such triples violating rad(abc) ≥ c^{1-ε} for any fixed ε > 0. The argument combines upper bounds on the density of integer points on certain high-dimensional varieties, derived via the geometry of numbers and Fourier analysis.

Significance. If the stated density bounds hold with positive exponents, the result supplies the first explicit power-saving estimate on the exceptional set in the abc conjecture. This is a quantitative strengthening of the known finiteness statements and demonstrates that the methods of geometry of numbers and Fourier analysis can be combined to produce a saving; the paper does not claim the full conjecture but a weaker, verifiable statement about exceptions.

minor comments (3)

The abstract states the existence of a power-saving bound but does not record the explicit exponent or the implied constant; adding this (even as O(X^θ) with θ < 1) would make the claim immediately verifiable from the opening paragraph.
The handling of the coprimality condition gcd(a,b,c)=1 is mentioned in the abstract but not expanded in the provided description; a short paragraph clarifying how the coprimality is preserved or removed in the density estimates would improve readability.
Notation for the exceptional set (e.g., whether it is counted by max(a,b,c) ≤ X or by c ≤ X) should be fixed consistently from the introduction onward.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for the recommendation of minor revision. The report accurately captures that our work establishes the first explicit power-saving bound on the exceptional set for the abc conjecture via density estimates on high-dimensional varieties.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation obtains a power-saving bound on the exceptional set via upper bounds on integer points on high-dimensional varieties, using geometry of numbers and Fourier analysis. These are presented as independent external tools rather than self-derived or fitted inputs. No load-bearing self-citations, self-definitional steps, or reductions of predictions to fitted parameters are indicated in the abstract or described methods. The central claim remains independent of the result itself and relies on standard analytic techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard facts about the radical function, coprimality, and the geometry of integer points on varieties; no free parameters or invented entities are visible in the abstract.

axioms (2)

standard math The radical function rad(n) is the product of distinct prime factors of n.
Invoked in the definition of the abc quality and exceptional set.
domain assumption Standard upper bounds on the number of integer points on varieties can be obtained via geometry of numbers and Fourier analysis.
Central to the proof strategy described in the abstract.

pith-pipeline@v0.9.0 · 5626 in / 1202 out tokens · 31363 ms · 2026-05-23T19:19:44.231989+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We obtain a power-saving bound on the size of the exceptional set of triples... upper bounds for the density of integer points on certain high-dimensional varieties, coming from the geometry of numbers and from Fourier analysis.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The proof is based on a combination of bounds for the density of integer points on varieties, coming from the determinant method, Thue equations, geometry of numbers, and Fourier analysis.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Bloom and J.D

T.F. Bloom and J.D. Lichtman, The Bombieri–Pila determinant method.Preprint, 2023. (arXiv:2312.12890)

work page arXiv 2023
[2]

Bombieri and J

E. Bombieri and J. Pila, The number of integral points on arcs and ovals. Duke Math. J. 59 (1989), 337–357

work page 1989
[3]

Bombieri and W.M

E. Bombieri and W.M. Schmidt, On Thue’s equation. Invent. Math. 88 (1987), 69–81

work page 1987
[4]

de Bruijn, On the number of integers ⩽ x whose prime factors divide n

N.G. de Bruijn, On the number of integers ⩽ x whose prime factors divide n. Illinois J. Math. 6 (1962), 137–141

work page 1962
[5]

Darmon and A

H. Darmon and A. Granville, On the equations zm = F (x, y) and Axp + By q = Cz r. Bull. London Math. Soc. 27 (1995), 513–543

work page 1995
[6]

Heath-Brown, Diophantine approximation with square-free numbers

D.R. Heath-Brown, Diophantine approximation with square-free numbers. Math. Zeit. 187 (1984), 335–344

work page 1984
[7]

Heath-Brown, The density of rational points on cubic surfaces

D.R. Heath-Brown, The density of rational points on cubic surfaces. Acta Arith. 79 (1997), 17–30

work page 1997
[8]

Heath-Brown, Counting rational points on algebraic varieties

D.R. Heath-Brown, Counting rational points on algebraic varieties. Analytic number theory, 51–95, Lecture Notes in Math. 1891, Springer-Verlag, 2006

work page 2006
[9]

Kane, On the number of ABC solutions with restricted radical sizes

D. Kane, On the number of ABC solutions with restricted radical sizes. J. Number Theory 154 (2015), 32–43

work page 2015
[10]

Mazur, Questions about powers of numbers

B. Mazur, Questions about powers of numbers. Notices Amer. Math. Soc. 47 (2000), 195–202

work page 2000
[11]

Pasten, The largest prime factor of n2 + 1 and improvements on subexponential ABC

H. Pasten, The largest prime factor of n2 + 1 and improvements on subexponential ABC. Invent. Math. 236 (2024), 373–385

work page 2024
[12]

Stewart and K.R

C.L. Stewart and K.R. Yu, On the abc conjecture. II. Duke Math. J. 108 (2001), 169–181. IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria Email address: tdb@ist.ac.at Department of Mathematics, Stanford University, 450 Jane Stanford W ay, Stanford, CA 94305- 2125, USA Email address: j.d.lichtman@stanford.edu Department of Mathematics and Statistics, ...

work page 2001

[1] [1]

Bloom and J.D

T.F. Bloom and J.D. Lichtman, The Bombieri–Pila determinant method.Preprint, 2023. (arXiv:2312.12890)

work page arXiv 2023

[2] [2]

Bombieri and J

E. Bombieri and J. Pila, The number of integral points on arcs and ovals. Duke Math. J. 59 (1989), 337–357

work page 1989

[3] [3]

Bombieri and W.M

E. Bombieri and W.M. Schmidt, On Thue’s equation. Invent. Math. 88 (1987), 69–81

work page 1987

[4] [4]

de Bruijn, On the number of integers ⩽ x whose prime factors divide n

N.G. de Bruijn, On the number of integers ⩽ x whose prime factors divide n. Illinois J. Math. 6 (1962), 137–141

work page 1962

[5] [5]

Darmon and A

H. Darmon and A. Granville, On the equations zm = F (x, y) and Axp + By q = Cz r. Bull. London Math. Soc. 27 (1995), 513–543

work page 1995

[6] [6]

Heath-Brown, Diophantine approximation with square-free numbers

D.R. Heath-Brown, Diophantine approximation with square-free numbers. Math. Zeit. 187 (1984), 335–344

work page 1984

[7] [7]

Heath-Brown, The density of rational points on cubic surfaces

D.R. Heath-Brown, The density of rational points on cubic surfaces. Acta Arith. 79 (1997), 17–30

work page 1997

[8] [8]

Heath-Brown, Counting rational points on algebraic varieties

D.R. Heath-Brown, Counting rational points on algebraic varieties. Analytic number theory, 51–95, Lecture Notes in Math. 1891, Springer-Verlag, 2006

work page 2006

[9] [9]

Kane, On the number of ABC solutions with restricted radical sizes

D. Kane, On the number of ABC solutions with restricted radical sizes. J. Number Theory 154 (2015), 32–43

work page 2015

[10] [10]

Mazur, Questions about powers of numbers

B. Mazur, Questions about powers of numbers. Notices Amer. Math. Soc. 47 (2000), 195–202

work page 2000

[11] [11]

Pasten, The largest prime factor of n2 + 1 and improvements on subexponential ABC

H. Pasten, The largest prime factor of n2 + 1 and improvements on subexponential ABC. Invent. Math. 236 (2024), 373–385

work page 2024

[12] [12]

Stewart and K.R

C.L. Stewart and K.R. Yu, On the abc conjecture. II. Duke Math. J. 108 (2001), 169–181. IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria Email address: tdb@ist.ac.at Department of Mathematics, Stanford University, 450 Jane Stanford W ay, Stanford, CA 94305- 2125, USA Email address: j.d.lichtman@stanford.edu Department of Mathematics and Statistics, ...

work page 2001