The exceptional set in the abc conjecture

Christian Bernert

REVIEW

The exceptional set in the abc conjecture

Christian Bernert

Number Theory · Submitted 16 Jun 2025 · 2 references

Pith's verdict: Small real improvement to the abc exceptional-set bound, shorter proof, but the key reduction is quoted from BLT24 and needs to be made explicit before the theorem is checkable.

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arXiv ID2506.13364 (version 1)

Pith IDGGDYALH4, Pith's permanent record of this paper and its review

CitationChristian Bernert. "The exceptional set in the abc conjecture." arXiv:2506.13364 (2025). https://pith.science/paper/GGDYALH4

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Editor's note

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Colleague,

The useful thing to know: Bernert gets a real but small improvement over BLT24—N_1(X) goes from X^{0.66+ε} to X^{0.65+ε}—and does it in four pages using the two simplest bounds. The endgame is a neat one-page optimization. If you work in this corner, worth a look.

The new thing is the exponent (23λ+3)/40, which strictly improves the BLT24 bound for λ∈(9/11,2). The propositions are explicitly variants of BLT24's Prop 3.1 and 3.2; the fourth-moment proof is compressed but looks sound. The divisor-bounded steps are standard. No fitted parameters, no self-citation, no invented entities. The arithmetic of the final contradiction checks out.

The soft spot is the reduction in §2. It quotes BLT24 Section 2 in one sentence and moves from N_λ to a uniform bound on B_d(c,X,Y,Z). As written, the paper does not state the hypotheses of that reduction, nor show how the sum over c_i and dyadic boxes is absorbed. The stress-test note's worry about the c_i in Prop 2 is mostly answered by c_i≥1, since the max including c_i is at least the max without them, so ≫ X. But the aggregation point is real: a uniform B_d bound implies N_λ only if the number of configurations is X^{o(1)}, and that is not argued here. I suspect it is true—a divisor bound should kill the choices—but the paper has to say it. A referee with BLT24 open can settle this in an afternoon.

Also minor: Theorem 1 states λ∈(0,2), but the proof only runs for λ>9/11. For the smaller λ the trivial bound is stronger, so the theorem still holds, but the proof should say so instead of silently fixing a δ that doesn't exist.

Verdict: this deserves a serious referee, not a desk reject. The improvement is marginal but honest, the method gets shorter, and the missing pieces are patchable rather than fundamental. I would condition acceptance on making §2 precise and adding the λ≤9/11 sentence. For the right specialist, cite-worthy.

Referee report

Major objections (3)

[§2] The reduction from N_λ(X) to the uniform bound on B_d(c,X,Y,Z) is asserted, not proved: the paper says only 'By the arguments from [BLT24, Section 2] (and using the monotonicity of our bound), it suffices...' and then states a sufficient condition. The manuscript does not state the hypotheses of that reduction, does not show that N_λ is dominated by a single B_d or by a sum with X^{o(1)} terms, and does not explain how the many possible c_i and dyadic configurations are aggregated. Since Theorem 1 depends entirely on this reduction, the central claim is not checkable from the paper alone.
[§2–§3, Proposition 2] The size hypotheses used for the reduction do not involve the integers c_1,c_2,c_3: they only impose ∏_i(X_iY_iZ_i) ∼ X^λ and max(∏_i X_i^i, ∏_i Y_i^i, ∏_i Z_i^i) ∼ X. In the proof of Proposition 2, however, the geometry-of-numbers bound requires max(|a_1|X_1, |a_2|Y_1, |a_3|Z_1) ≫ X, and with a_1 = c_1∏_{i≥2} x_i^i etc. this quantity is approximately max(c_1∏X_i^i, c_2∏Y_i^i, c_3∏Z_i^i). For arbitrary positive integers c_i, the printed hypotheses do not imply that the latter is ≫ X. The manuscript therefore needs to either incorporate the c_i into the size conditions or prove that the reduction can be arranged so that c_i are bounded or otherwise controlled.
[§4] The endgame uses the lower bound X^{5λ−3} ≪ ∏_i P_i^{5−i} without proof or citation. This inequality is essential for the contradiction, and it is not immediate from the stated hypotheses ∏ P_i ∼ X^λ and max(∏X_i^i,∏Y_i^i,∏Z_i^i) ∼ X. The author should provide a derivation or a precise reference to where this appears in [BLT24].

Minor objections (4)

[§2] The phrase 'using the monotonicity of our bound' is vague; please specify which monotonicity is meant and how it enters the reduction.
[§3, Proposition 3 proof] The summation notation in the Cauchy–Schwarz step, '∑_{x_i∼X_i, x'_j∼X_j}', is ambiguous: it should explicitly indicate that x_i ranges over all i (including x_j) together with a second variable x'_j, or else the displayed expression does not match the subsequent counting argument.
[§1] The sentence 'N_λ(X)=o(X^{2λ/3}) for all λ∈(9/11,3)' mixes the range of Theorem 1 (λ<2) with the known range of Kane's bound (λ≥2); the statement is true, but the wording should clarify that the λ∈[2,3) part is from [Kan15], not from Theorem 1.
[§4, Remark] The remark sketches numerical improvements without proofs; if these claims are intended to be part of the paper's contributions, they should be labeled as conjectural or backed by at least a sketch of the modified inequalities.

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The paper

The paper claims

The paper proves that the number of primitive triples with a+b=c, c≤X, and rad(abc)<c is at most X^{0.65+ε}, improving the previous X^{0.66+ε} bound.

The reading

This paper improves the known upper bound on the size of the exceptional set in the abc conjecture. The exceptional set consists of primitive triples (a,b,c) with a+b=c, c≤X, and radical rad(abc), the product of the distinct prime factors, smaller than c^λ; the paper proves that for each fixed λ∈(0,2) there are at most $X^{{(23λ+3)/40+ε}}$ such triples. For λ=1 this gives N_1(X)≪$X^{{0.65+ε}}$, a small but concrete improvement over the earlier $X^{{0.66+ε}}$. The proof is short and uses only the simplest forms of two known techniques, a geometry-of-numbers bound and a Fourier-analytic moment bound, together with a product inequality among the box sizes. A refined variant mentioned in the paper would push λ=1 to $X^{{8/13+ε}}$.

What carries the argument

The central object is the auxiliary count B_d(c,X,Y,Z), defined as the number of primitive solutions of c_1∏_{i=1}^d x_i^i + c_2∏_{i=1}^d y_i^i = c_3∏_{i=1}^d z_i^i with variables in dyadic boxes x_i∼X_i, y_i∼Y_i, z_i∼Z_i. The reduction from the abc exceptional count to these counts is quoted from [BLT24, Section 2]. The proof needs two bounds: Proposition 2, a geometry-of-numbers bound giving B_d ≪ X^λ/P_1 + $X^{{λ−1}}$, and Proposition 3, a symmetrized Fourier bound giving B_d ≪ $X^{{2λ/3+ε}}$/$P_j^{{1/6}}$ for any j≥2, where P_i=X_iY_iZ_i. The endgame combines the constraints these bounds force on the P_i with the product inequality $X^{{5λ−3}}$ ≪ ∏_i $P_i^{{5−i}}$ to reach the claimed exponent.

Extended reading notes

Core claim

Theorem 1 of the paper states that for every λ∈(0,2), N_λ(X), the number of primitive solutions of a+b=c with c≤X and rad(abc)<c^λ, is O_ε($X^{{(23λ+3)/40+ε}}$). In particular N_1(X)≪$X^{{0.65+ε}}$, which is the sharpest known exponent at λ=1 and improves the earlier 0.66. The theorem also gives N_λ(X)=o($X^{{2λ/3}}$) for every λ∈(9/11,3), so the simple bound with exponent 2λ/3 is not optimal across that range. The author presents the result as a demonstration that two of the four methods used in the earlier treatment already suffice to beat the previous record, and sketches a refined version that would lower the λ=1 exponent further.

Load-bearing premise

The argument rests on a reduction, quoted without proof, that turns each abc exception into one of the auxiliary product-equation configurations; if that reduction secretly fails for some parameter choices, the main theorem does not follow.

Editorial extensions

If this is right

For λ=1, the count of primitive triples with rad(abc)<c and c≤X is bounded by X^{0.65+ε}, improving the previous X^{0.66+ε}.
The bound N_λ(X)=o(X^{2λ/3}) holds for every λ∈(9/11,3), showing the simple exponent 2λ/3 is not sharp in that range.
For λ∈(0,2), the uniform bound X^{(23λ+3)/40+ε} applies, while for λ≥2 the known X^{λ−1+ε} bound covers the remaining range.
The refined sketch would give N_1(X)≪X^{8/13+ε}, a further reduction below 0.6154.
Because only the two simplest bounds are needed, the argument avoids the computational optimization used in the earlier treatment.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same two-bound-plus-product endgame might transfer to other Diophantine counting problems where a product structure and a radical-size condition appear.
The fourth-moment Fourier estimate could be replaced by higher moments, which would most likely lower the exponent further; the paper's final remark already points in this direction.
Until the quoted reduction is written out in full, the theorem's range of validity is tied to the correctness of that reduction; making it explicit would be a useful check.
The new exponent is still very far from the abc-conjecture prediction that N_λ(X) is finite for λ<1, so the result is best read as an incremental measurement of how far power-saving methods currently reach.

Original abstract

We study the size of the exceptional set in the $abc$ conjecture, improving on and simplifying work of Browning, Lichtman and Ter\"av\"ainen.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 1,620 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.91 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We study the size of the exceptional set in the $abc$ conjecture, improving on and simplifying work of Browning, Lichtman and Ter\"av\"ainen.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study the size of the exceptional set in the $abc$ conjecture, improving on and simplifying work of Browning, Lichtman and Ter\"av\"ainen.	contribution 1.00	0.86	0.33	asserted 0.97

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.98.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
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P abstract

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S sections
T body sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

How the literature uses it

What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 2 sentences without a confident label

Uses the method or tool of 1 paper

It is a symmetrized version of [ref] .

Bounds on the exceptional set in the $abc$ conjecture · The two bounds · 0.91

Compares with 1 paper

This improves on the trivial bound $N_λ(X) ≪X^2λ/3+ε$ (see [ref] ).

Bounds on the exceptional set in the $abc$ conjecture · Introduction · 0.83

Discussion

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[1] [1]

Tim Browning, Jared Duker Lichtman, and Joni Teräväinen. Bounds on the exceptional set in the abc conjecture, arXiv:2410.12234, 2024

View on Pith Pith review arXiv 2024

[2] [2]

Daniel M. Kane. On the number of ABC solutions with restricted radical sizes. J. Number Theory , 154:32--43, 2015

View on Pith 2015