REVIEW
Roth's theorem on weighted projective lines
Number Theory · Submitted 13 Oct 2026 · 35 references
Pith's verdict: Clean, honest paper extending Roth-type results to weighted projective lines; main result conditional on abc, but unconditional partial results give it solid footing.
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Referee report
The paper
The paper claims
Under the abc conjecture, integral points on the weighted projective line attain the sharper approximation exponent q0+q1, and this exponent is optimal for weights (2,3).
The reading
What carries the argument
The weighted Veronese map Φ: $P^{1}$_q → $P^{1}$, [x0:x1]_q ↦ [$x0^{{q1}}$: $x1^{{q0}}$], which identifies the weighted projective line with $P^{1}$ and satisfies h(Φ(P)) = L s(P). For integral points (coprime integer coordinates), Φ(P) gives a fraction whose numerator is a q1-th power and denominator a q0-th power, so rad(u)rad(v) ≤ |x|$_q^{{q0+q1}}$. Under the abc conjecture, this radical bound, combined with Belyi maps transferring the target to a rational point, forces the smaller exponent q0+q1 in the Roth inequality.
Extended reading notes
Core claim
The central discovery is a sharp Roth-type inequality for integral points on the weighted projective line: for any algebraic target α with Φ(α) not on the coordinate axes, the weighted chordal distance δ satisfies -log δ(P,α) ≤ (q0+q1+ε)s(P) + O(1) for all but finitely many integral points P, assuming the abc conjecture. The exponent q0+q1 is optimal when q=(2,3) and Φ(α)=[1:1], where it reduces to Hall's conjecture for $x^{3}$ - $y^{2}$. The proof combines the weighted Veronese map (which expresses the weighted height as 1/L of the ordinary height) with Belyi's theorem and the abc conjecture: integrality forces the numerator and denominator of Φ(P) to be perfect powers, so their radicals are small, and abc controls the size of radical(G(u,v)). The paper also introduces a weighted abc conjecture and weighted Belyi maps to show that the full abc conjecture is equivalent to the weighted conjecture for a single signature.
Load-bearing premise
The strongest results for number fields (Theorem 5.7) assume the abc conjecture, which remains unproven. The weights q0,q1 must be coprime, and the target must avoid the two coordinate axes.
Editorial extensions
If this is right
- For weights (2,3) and target [1:1], the inequality specializes to the coprime Hall-type bound |x^3-y^2| ≫_ε |(x,y)|^{5-ε}, which is sharp (Proposition 5.18).
- Weighted Thue equations F(x,y)=m with κL > q0+q1 have only finitely many coprime integer solutions under abc (Corollary 7.6).
- The abc conjecture is equivalent to the weighted abc conjecture wABC(3,2;4e,-1) for e∈{1,3,9,27} (Proposition 5.15).
- Unconditionally, the exponent q0+q1 holds over characteristic-zero function fields for constant targets and polynomial approximants (Proposition 5.19).
- For almost every real target, the inequality holds unconditionally over Q (Proposition 5.20).
Reading between the lines
- The result suggests that the gap between 2L and q0+q1 originates entirely from the small radicals of perfect powers; a similar mechanism may improve Roth bounds for integral points on other weighted projective varieties.
- The equivalence of full abc to a finite set of radical inequalities of signature (3,2) indicates that the integral-point conjecture for (2,3) and four rational targets is a 'test case' for abc.
- The failure of direct weighted index rescaling (Section 8) implies that any unconditional approach to the sharp exponent must incorporate arithmetic information, such as radical bounds, rather than purely geometric index arguments.
Original abstract
Let ${\mathbb P}^1_{\mathfrak q}$ be the weighted projective line with coprime weights ${\mathfrak q}=(q_0,q_1)$ and $L=q_0q_1$. The weighted Veronese map $\Phi\colon {\mathbb P}^1_{\mathfrak q}\to {\mathbb P}^1$ satisfies $h\circ\Phi=L\,{\mathfrak s}$. Pulling back Roth's theorem gives the optimal exponent $2L$ for the full rational orbit set. For points represented by coprime integral coordinates, counting suggests the smaller exponent $q_0+q_1$. We formulate this as a conjecture and prove an $S$-adic radical form under the $abc$ conjecture. For ${\mathfrak q}=(2,3)$ the resulting inequality at the target $[1:1]$ is the coprime Hall-type inequality, and Danilov--Elkies examples show that the exponent $5$ is sharp. We also prove the expected exponent unconditionally over characteristic-zero function fields for constant targets and polynomial approximants, and for almost every real target. A weighted $abc$ formulation is introduced; in signature $(3,2)$ finitely many coefficient pairs are equivalent to $abc$, and weighted Belyi maps transfer this input to arbitrary algebraic targets. Applications include a pullback form of Schmidt's subspace theorem, weighted Thue inequalities, and a counterexample to a naive weighted rescaling of Roth's index construction.
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.
The abstract, sentence by sentence
Let ${\mathbb P}^1_{\mathfrak q}$ be the weighted projective line with coprime weights ${\mathfrak q}=(q_0,q_1)$ and $L=q_0q_1$. The weighted Veronese map $\Phi\colon {\mathbb P}^1_{\mathfrak q}\to {\mathbb P}^1$ satisfies $h\circ\Phi=L\,{\mathfrak s}$. Pulling back Roth's theorem gives the optimal exponent $2L$ for the full rational orbit set. For points represented by coprime integral coordinates, counting suggests the smaller exponent $q_0+q_1$. We formulate this as a conjecture and prove an $S$-adic radical form under the $abc$ conjecture. For ${\mathfrak q}=(2,3)$ the resulting inequality at the target $[1:1]$ is the coprime Hall-type inequality, and Danilov--Elkies examples show that the exponent $5$ is sharp. We also prove the expected exponent unconditionally over characteristic-zero function fields for constant targets and polynomial approximants, and for almost every real target. A weighted $abc$ formulation is introduced; in signature $(3,2)$ finitely many coefficient pairs are equivalent to $abc$, and weighted Belyi maps transfer this input to arbitrary algebraic targets. Applications include a pullback form of Schmidt's subspace theorem, weighted Thue inequalities, and a counterexample to a naive weighted rescaling of Roth's index construction.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let ${\mathbb P}^1_{\mathfrak q}$ be the weighted projective line with coprime weights ${\mathfrak q}=(q_0,q_1)$ and $L=q_0q_1$. | background 0.99 | 0.14 | 0.02 | asserted 0.87 |
| s1 | The weighted Veronese map $\Phi\colon {\mathbb P}^1_{\mathfrak q}\to {\mathbb P}^1$ satisfies $h\circ\Phi=L\,{\mathfrak s}$. | background 0.51 | 0.35 | 0.06 | asserted 1.00 |
| s2 | Pulling back Roth's theorem gives the optimal exponent $2L$ for the full rational orbit set. | result 0.89 | 0.69 | 0.33 | asserted 1.00 |
| s3 | For points represented by coprime integral coordinates, counting suggests the smaller exponent $q_0+q_1$. | result 0.29 | 0.42 | 0.09 | hedged possibility 0.98 |
| s4 | We formulate this as a conjecture and prove an $S$-adic radical form under the $abc$ conjecture. | contribution 0.88 | 0.93 | 0.21 | asserted 0.82 |
| s5 | For ${\mathfrak q}=(2,3)$ the resulting inequality at the target $[1:1]$ is the coprime Hall-type inequality, and Danilov--Elkies examples show that the exponent $5$ is sharp. | result 0.84 | 0.30 | 0.10 | asserted 1.00 |
| s6 | We also prove the expected exponent unconditionally over characteristic-zero function fields for constant targets and polynomial approximants, and for almost every real target. | result 0.92 | 0.94 | 0.15 | asserted 1.00 |
| s7 | A weighted $abc$ formulation is introduced; in signature $(3,2)$ finitely many coefficient pairs are equivalent to $abc$, and weighted Belyi maps transfer this input to arbitrary algebraic targets. | contribution 0.87 | 0.85 | 0.07 | asserted 1.00 |
| s8 | Applications include a pullback form of Schmidt's subspace theorem, weighted Thue inequalities, and a counterexample to a naive weighted rescaling of Roth's index construction. | result 0.50 | 0.89 | 0.10 | asserted 1.00 |
What was examined, and what was not
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References
Discussion
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