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Multiplicative dependence in the sumset of multiplicative groups

T0 review · 2 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The only infinite family of multiplicatively dependent sums from two almost disjoint multiplicative groups comes from multiplying both summands of one sum by a common root of unity.

desk verdict The m=2 theorem is solid and citable; the abc-conditional m=3 classification is a real advance, but the proof of Theorem 7.3 has a gap in the b+c=0 sub-case that needs fixing. read the letter →

arxiv 2607.28857 v1 pith:7I2CSU33 submitted 2026-07-30 math.NT

Yuri Bilu , Florian Luca This is my paper · ORCID
classification math.NT MSC 11D6111J8611R27
keywords multiplicativedependencesumsetfinitelygeneratedgroupsalgebraicnumbersrootofunityabcconjectureexponentialDiophantineequationsheights
topics The abc Conjecture
open problems The abc Conjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies when two sums from the sumset of two finitely generated multiplicative groups of algebraic numbers can be multiplicatively dependent. The main theorem states that, if the two groups intersect only in a finite group, then with finitely many exceptions the only way this happens is that the two sums are related by multiplying both summands by the same root of unity: x1/x2 = y1/y2 is a root of unity. For three sums, the paper shows — assuming the weak abc conjecture — that only two infinite families occur, up to permutation and root-of-unity twists: the difference-of-squares family and a reciprocal family. These results matter because they pin down when multiplicative dependence can occur in the sumset, a question that arises naturally in exponential Diophantine equations. The paper also proposes a conjectural description of m sums for arbitrary m.

What carries the argument

Almost disjointness yields a height comparison: for x∈Γ and y∈Δ, h(xy) ≫ h(x)+h(y). This prevents height cancellation in products and forces a multiplicative dependence between two sums to mean either a perfect power in Γ+Δ or equality of the sums up to a root of unity. For three sums, the proof reduces to a structural theorem classifying triples (x,y,z) in one group with x−1, y−1, z−1 multiplicatively dependent modulo the group; that theorem uses linear-forms-in-logarithms bounds, a logarithmic-gcd estimate, the primitive-divisor theorem, and the weak abc conjecture.

What would settle it

Take two almost disjoint rank-1 groups, for instance Γ=⟨2⟩ and Δ=⟨3⟩, and determine whether the equation 2^a+3^b = z^n has infinitely many solutions with n≥2; the paper's Theorem 4.1 predicts only finitely many, so one infinite parametric family would refute it. For Theorem 1.2 itself, one would need an infinite sequence of dependent pairs (x1,y1),(x2,y2) from almost disjoint finitely generated groups with x1/x2 not a root of unity; a single example cannot refute the theorem because it allows finitely many exceptions.

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Extended reading notes

Core claim

The central claim is Theorem 1.2: if Γ and Δ are almost disjoint finitely generated multiplicative groups of algebraic numbers, and x1+y1 and x2+y2 are non-zero and multiplicatively dependent with x1,x2∈Γ, y1,y2∈Δ, then, with finitely many exceptions, x1/x2 = y1/y2 is a root of unity. The proof splits by the exponent of the dependence: if the common power has exponent at least 2, the sum is a perfect power and the finiteness of perfect powers in Γ+Δ bounds heights; if the exponent is 1, the sums are either equal up to a root of unity or their product is a root of unity, each handled by classical unit-equation finiteness. The theorem is non-effective in general, but effective when both groups

Load-bearing premise

The main theorem collapses if Γ and Δ meet in an infinite group: Example 1.3 shows that Γ=⟨−1,3⟩ and Δ=⟨2,3⟩ give every power of 3 in Γ+Δ, so multiplicative dependence is abundant. The three-sum classification additionally rests on the unproved weak abc conjecture for the number field K.

Editorial extensions

If this is right

  • If Γ and Δ are almost disjoint, any two multiplicatively dependent non-zero sums from Γ+Δ must, with finitely many exceptions, have x1/x2 = y1/y2 a root of unity.
  • The sumset Γ+Δ contains only finitely many perfect powers in any given number field, with an effective height bound.
  • For three sums, assuming weak abc, any minimally dependent triple is, up to permutation and torsion, either of the form (x−y, x+y, x²−y²) or of the reciprocal form (x1+y1, x2+y2, x2⁻¹+y2⁻¹) with x1+y1 multiplicatively dependent on x2y2.
  • The rank-1 case of Theorem 1.2 is effective, giving explicit height bounds for the exceptional pairs.

Reading between the lines

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

  • The non-effectiveness of the general m=2 statement seems tied to the non-effective finiteness theorem used to rule out unequal ratios; any effective generalization would likely need a new height-matching argument or a restriction to low rank, as the rank-1 case illustrates.
  • If the conjectural description for arbitrary m (Conjecture 1.6) is correct, then all minimally dependent m-tuples are built by concatenating canonical pairs (from differences of powers) and twisted pairs (from reciprocal/inverse pairs), yielding a complete structural description of multiplicative dependence in sumsets of almost disjoint groups.
  • One could test the conditional three-sum classification in the rank-1 setting, where the proof's non-effective ingredients are replaced by Baker-type explicit bounds, giving a fully effective version of Theorem 1.5 for rank-1 groups.
  • The weak abc assumption enters only in bounding conductors; for a specific number field K for which Conjecture 3.9 is verified, Theorem 1.5 would become unconditional for that field.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies multiplicative dependence of non-zero sums x_i + y_i with x_i in a finitely generated multiplicative group Gamma and y_i in an almost disjoint finitely generated group Delta of algebraic numbers. Theorem 1.2 gives an unconditional classification for m=2: apart from finitely many exceptions, x_1/x_2 = y_1/y_2 is a root of unity. The proof reduces to perfect powers in sumsets (Theorem 4.1) and to S-unit equations, with the unavoidable non-effectiveness coming from the subspace theorem. For m=3, Theorem 1.5 gives, conditionally on the weak abc-conjecture, a classification up to finitely many exceptions: the only infinite families are the two described in (1.1) and (1.2), up to permutation and root-of-unity twists. The technical core is Theorem 7.3, a conditional statement about three elements x,y,z of a finitely generated group Lambda for which x-1, y-1, z-1 are minimally multiplicatively dependent modulo Lambda. The final section derives Theorem 1.5 from Theorem 7.3. A general conjecture, Conjecture 1.6, is also proposed.

Significance. If correct, Theorem 1.2 is a clean and natural contribution: it identifies the only infinite family of multiplicatively dependent pairs in the sumset of almost disjoint multiplicative groups, and it is unconditional apart from the inherent non-effectiveness. The conditional m=3 classification is strong and well-motivated, and the paper is honest about its dependence on the weak abc-conjecture. The structure is good, and the paper gives explicit examples showing that the hypotheses are sharp. However, as demonstrated below, the proof of Theorem 7.3 contains a genuine gap in the analysis of a six-term unit equation, and since Theorem 1.5 is deduced from Theorem 7.3, this gap is load-bearing for the conditional classification. The unconditional Theorem 1.2 appears sound.

major comments (2)
  1. [§7.4.2, equation (7.32), case b+c=0] The assertion that, because y^{|r|} != 1, a proper zero sub-sum must involve only the first four coordinates of (7.32) is unjustified and in fact false. A zero sub-sum may include the fifth or sixth coordinate. For example, with x=2, y=3, gamma=1, r=1, the sub-sum xy - y - gamma y^r = 6-3-3 = 0 vanishes, while no proper subset of {xy, -x, -y, 1} sums to zero. The case analysis that follows checks only two- and three-term sums among the first four coordinates; all zero sub-sums involving -gamma y^{|r|} or gamma are omitted. Some of these omitted cases may be excluded by Corollary 3.7 or Theorem 3.6, but the manuscript does not perform that exclusion. Since Theorem 1.5 is derived from Theorem 7.3, this gap must be repaired.
  2. [§7.4.2, three-term sums after (7.32)] In the same paragraph, the list of three-term sums contains a sign/justification error. With coordinates (xy, -x, -y, 1), the relevant three-term sums include xy-x-y, xy-x+1, xy-y+1, and 1-x-y; the paper's '-x-y-1' is not the correct expression. More importantly, the vanishing of 1-x-y (i.e., x+y=1) does not imply that x-1 and y-1 are S-units; for instance, x=2, y=-1. The finiteness of S-unit solutions to x+y=1 follows from Theorem 3.6, but the printed argument uses a false implication. This is local and repairable, but it must be corrected.
minor comments (4)
  1. [§5.3.6] The sentence 'Since groups Gamma and Delta are almost disjoint, gamma and delta must be algebraically independent' should read 'multiplicatively independent'. Two algebraic numbers cannot be algebraically independent over Q in the usual sense. The intended argument works once 'multiplicatively' is substituted.
  2. [§7.4.2, equation (7.31)] In (7.31), gamma should be an S-unit, not merely an element of O_S; the derivation from (7.29) gives gamma in O_S^times. The subsequent use of (7.32) as a six-term S-unit equation requires this.
  3. [§8.2] The final line of the paper says 'Theorem 7.3 is proved'; this should be 'Theorem 1.5 is proved'.
  4. [§3.4, Conjectures 3.8 and 3.9] The conductor terms appear to be mis-transcribed: the standard weak abc inequality should involve cond(x) + cond(x^{-1}) + cond(x-1). The displayed text repeats cond(x-1) twice. If the literal statement is intended, it is weaker than the form used in Proposition 7.20, where the full sum is bounded via Proposition 7.19. Please correct the typo or clarify.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity: central results derive from external tools; self-citations are background only.

full rationale

Walking the derivation chain, I find no step in which a claimed prediction or theorem is equivalent by construction to its own inputs. Theorem 1.2 is proved from external machinery: Baker's inequality (Theorem 3.4), the Evertse-Schlickewei-van der Poorten subspace-technique theorem (Theorem 5.1), the binary unit equation bound (Theorem 3.6), and the perfect-power-in-sumset result (Theorem 4.1). None of these presupposes the conclusion x1/x2 = y1/y2 is a root of unity. Theorem 1.5 is conditional on Conjecture 3.9, an external unproved statement; relying on abc is a genuine dependency, not a circularity. The reduction in Section 8 sets z_k = -x_k/y_k and applies Theorem 7.3; the two statements are independent reformulations, and the proof does not assume Theorem 1.5 in proving Theorem 7.3 or vice versa. Self-citations [3], [4], [5] occur only for standard background facts (Dirichlet's height lower bound, Schinzel's primitive divisor theorem) and are not used as the load-bearing justification for the paper's new classification claims. The skeptical concern about §7.4.2 — that the six-term unit equation analysis omits sub-sums involving the last two coordinates — is a possible correctness gap in the proof of Theorem 7.3, but it is not a circularity: the missing case analysis is not an identity between the theorem's hypothesis and conclusion. Overall, the central derivations are self-contained against external benchmarks; the only reason the score is not 0 is the presence of minor, non-load-bearing self-citations.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper is a theorem-proving work in Diophantine equations; no free parameters are fitted to data, and no new unobserved entities are introduced. The load-bearing assumptions are deep standard theorems (subspace theorem, Baker, primitive divisors, Corvaja–Zannier) and, for the m=3 results, the weak abc conjecture. All are clearly attributed.

assumptions (7)
  • domain assumption Weak abc conjecture for number fields (Conjecture 3.9): there exists κ>1 such that h(x) ≤ κ(cond(x)+cond(x^{-1})+cond(x-1)+1) for all x∈K with x≠0,1.
    Assumed for the m=3 classification (Theorem 1.5 and Theorem 7.3); stated explicitly in §3.4. It is an unproved conjecture, not a theorem.
  • standard math Evertse–Schlickewei–van der Poorten theorem (Theorem 5.1): the equation x1+...+xm=0 in a finitely generated multiplicative group has finitely many primitive solutions up to equivalence.
    Used in Propositions 5.2 and 7.2 and Section 8; a published theorem based on the subspace theorem.
  • standard math Baker's inequality (Theorem 3.4) and its p-adic analogue (Yu).
    Used throughout (Prop 4.3, Theorem 5.3, Corollary 3.5); a published theorem.
  • standard math Corvaja–Zannier logarithmic gcd bound (Theorem 7.6).
    Used in Proposition 7.20 to bound logarithmic gcds of x−1 and z−1; a published theorem.
  • standard math Schinzel's primitive divisor theorem (Theorem 7.12) and consequences (Prop 7.14).
    Used in Proposition 7.18; a published theorem.
  • standard math Effective bounds for superelliptic equations (Theorem 4.4, citing Bérczes–Bugeaud–Győry–Mello–Ostafe–Sha).
    Used in proof of Theorem 4.2 to bound heights of solutions to AX^3+1=BY^n; a published theorem.
  • standard math Northcott's theorem on finitely many algebraic numbers of bounded height.
    Used in Propositions 7.4 and 7.16 to build finitely generated group extensions; a standard theorem.

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Pith. "Pith review of Multiplicative dependence in the sumset of multiplicative groups." pith.science (2026). https://pith.science/paper/7I2CSU33

@misc{pith2026260728857,
  author       = {Pith},
  title        = {Pith review of: Multiplicative dependence in the sumset of multiplicative groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7I2CSU33}},
  note         = {Machine review of arXiv:2607.28857}
}
abstract

Let $\Gamma$ and $\Delta$ be finitely generated multiplicative groups of algebraic numbers such that $\Gamma\cap\Delta$ is a finite group. We show that, up to finitely many exceptions, non-zero sums $x_1+y_1$ and $x_2+y_2$, with $x_1, x_2\in \Gamma$ and $y_1,y_2\in \Delta$, are multiplicatively dependent only if $x_1/x_2=y_1/y_2$ is a root of unity. For $m\ge 3$, we discuss possible shapes of $m$ multiplicatively dependent sums $x_1+y_1, \ \ldots, \ x_m+y_m$ with $x_1, \ldots, x_m \in \Gamma$ and $y_1, \ldots, y_m \in \Delta$. For $m=3$ we classify such sums, up to finitely many exceptions, assuming the $abc$-conjecture.

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